Utilizing spatial statistical models to reduce data redundancy and entropy

ABSTRACT

A method, article comprising machine-readable instructions and apparatus that processes data systems for encoding, decoding, pattern recognition/matching and data generation is disclosed. State subsets of a data system are identified for the efficient processing of data based, at least in part, on the data system&#39;s systemic characteristics.

This application claims the benefit of priority to U.S. ProvisionalPatent Application Ser. No. 61/606,446 filed on Mar. 4, 2012. These andall other extrinsic materials discussed herein are incorporated byreference in their entirety. Where a definition or use of a term in anincorporated reference is inconsistent or contrary to the definition ofthat term provided herein, the definition of that term provided hereinapplies and the definition of that term in the reference does not apply.

FIELD OF THE INVENTION

The field of the inventive subject matter is data processing, whichincludes: statistical modeling, data encoding and decoding, datasimulation and branches of artificial intelligence, such as patternrecognition.

BACKGROUND

Patterns of data are described using models, which allow data processingmethods to remove information redundancy for lossless and lossy datacompression, and reduce the number of calculations required for patternrecognition, data generation and data encryption. Four basic datamodeling techniques exist in the known art, which are statisticalmodeling, dictionary coding, combinatorial coding and mathematicalfunctions.

Statistical modeling determines a probability for a state or symbolbased on a number of times the symbol occurs. The probabilities arerecorded in an index, which can then be accessed by a decoder todecipher the message. An encoder can generate a more efficient code byimplementing the model. It is why Morse code uses short codes for “A, E,I and U” and long codes for “X, Y, Z” or “0-9”, for the vowels in theEnglish alphabet are modeled with a higher probability than consonantsand numbers.

Assigning probability values also enables a pattern recognition methodto reduce a number of states to select from when matching or recognizingdata. For voice or speech recognition, the Hidden Markov model is used,which can use an index to assign probabilities to the possible outcomes.

The second technique typically used for modeling data is the dictionarycoder, which records patterns of strings by assigning a reference to thestring's position. To eliminate redundancy, the number of referencesmust be less than the number of possible patterns for a string, givenits length. The references substitute for the strings to create anencoded message. To reconstruct the message, the decoder reads thereference, looks it up in the dictionary and writes the correspondingstring.

The decoder must access the statistical index or dictionary to decodethe message. To allow for this access, the index/dictionary can beappended to the encoded message. Appending the index or dictionary maynot add much to the compressed code if a relatively short number ofstates are modeled. If the number of patterns in the index or dictionaryis too large, then any advantage gained by compressing the data can beeliminated after the statistical index or dictionary is appended.

In the known art, an adaptive index or dictionary can be used to solvethe problem of appending it to the encoded message. In adaptivemodeling, both the encoder and the decoder use the same statisticalmodel or dictionary at the start of the process. It then reads each newstring and updates the model as the data is being encoded or decoded.This helps to improve compression ratios, for the model is not addedwith the message.

Two main problems exist when using an adaptive index and dictionary. Thefirst is that it is relatively inefficient near the beginning of thedata stream. This is due to the fact that the encoder starts with usinga small number of patterns in the model. The smaller the number ofpatterns modeled, the less accurate the model is relative to the size ofthe message. Its efficiency may improve once the number of patternsincreases as the adaptive model grows. A more significant problem isthat, like the static index or dictionary, the adaptive model must beconstructed and stored in memory. Modeling more patterns can depletememory resources when the index or dictionary becomes too large. Morecalculations are also required to update the index/dictionary. The thirdproblem is that adaptive modeling is more computationally expensivecompared to static indexes or dictionaries because the model must beconstantly updated as the decoder decodes the message. For example,using adaptive compression with Huffman tree codes requires the dataprocessor to continuously update the nodes and branches of the code datatree as it encodes/decodes. For arithmetic coding, updating theprobabilities for each symbol pattern requires updating the counts forall the subsequent symbol patterns as well. This can take a considerableamount of time for the processor to calculate the probabilities for alarge index, for the number of possible patterns rises exponentiallywith each bit added. The adaptive technique can therefore slowproductivity of a device requiring frequent encoding/decoding of data,such as medical data, audio, video or any other data that requires rapidaccess. This can be especially problematic for mobile devices, whichtypically hold less memory and processing power than personal computersand database machines.

The third modeling technique involves combinatorial encoding. Asexemplified in U.S. Pat. No. 7,990,289 to Monro titled “CombinatorialCoding/Decoding for Electrical Computers and Digital Data ProcessingSystems” filed Jul. 12, 2007, combinatorial encoding counts a number oftimes a symbol appears in a sequence and generates a code describing itspattern of occurrences. This method can be effective for text documentswhere there is usually a statistical bias in the number of counts foreach symbol or, when the numbers of occurrences are predetermined orknown to the decoder. This statistical bias may be used in combinatorialcoding to compress data.

A problem with this method is that the effectiveness may lessen if thereis no statistical bias or, when the numbers of counts are relativelyequal and unknown. When the counts for each symbol reach parity, thenumber of combinations is at its highest, resulting in very littlecompression. Also, the number of occurrences of each symbol needs to be,like an index/dictionary, accessible to the decoder in order to decodethe encoded message describing the pattern of occurrences. Like theproblem associated with appending dictionary encoders, any compressiongained by encoding the pattern of occurrences can be nullified if anindex describing the number of occurrences for each symbol is too large.

The fourth modeling technique involves signal processing, which includesthe modeling of waveforms and time series data using mathematicalfunctions. Such methods usually extract three basic patterns of thedata; trends, cycles and white noise. Trends describe a gradual tendencyfor the signal to increase or decrease in time. Cycles describerepeating patterns in the data, such as frequencies. White noise isconsidered random-like data, which offers no discernible patterns. Thesethree pattern types can be calculated within a time, frequency ortime-frequency domain. Such pattern extraction techniques includeautocorrelation, Fourier analysis and wavelets. By using mathematicalfunctions to decipher patterns of a signal, the inverse of the functionscan either approximate the signal or reconstruct it. These models canthen be used for analyzing and forecasting stock prices and, for lossydata compression.

The problem associated with using mathematic functions as a model isthat the functions tend to identify only general properties of the data.Reconstructing finer details of the signal is computationally expensive.Secondly, they offer no known way of generating a probability for aunique data sequence deterministically, for probabilities are notincorporated into the calculation. Probability values are required tomeasure the information entropy of a sequence. Therefore, thesetechniques are generally used for approximating signals using lossy datacompression or forecasting; not for lossless data compression,particularly involving literal data such as text or machine code.

One may see that a fundamental problem for all of the four modelingtechniques is that they are all memory and computationally expensivewhenever their models describe the probabilities for large numbers ofstates, long sequences of data or, data with high entropy. Modeling alarge number of states increases the model's efficiency, but it alsotakes a toll on the data processor and memory. For example, with patternrecognition, the Hidden Markov model, coupled with a dynamic programmingtechnique, can be computationally intensive the more outcomes there areto solve for. The only way to reduce the computational complexity ofpattern recognition and data compression using the modeling techniquesin the known art is to reduce the number of patterns to model for.

Lossy encoders, however, attempt to encode the general meaning of amessage similarly to signal analysis. For example, JPEG compressionmethods analyze the brightness and chrominance of each pixel andattempts to find statistical redundancy for those values. Because humansperceive more detail in the brightness of an image rather than in itshues, the chrominance can be down sampled, which eliminates some of theinformation by using a statistical model tailored to the way humansperceive light.

Lossy encoding generally achieves higher compression ratios thanlossless methods because the finer details of the message are not asimportant as the message's broader values. The human visual perceptionis not based on a pixel by pixel scan of an image, but on a wide view ofthe image. This is also the case with sound and language. Humans canusually understand a sentence without requiring each letter or word tobe accurate. Most people will be able to understand the followingsentence in the English language, without all the letters or words beingaccurate: “I lov yu.”

The problem with lossy compression is that it sacrifices information inorder to encode or process the data. If too much information issacrificed, the image, video or sound quality is degraded, which isundesirable for many applications, such as high definition video oraudio. Even with lossy compression, the amount of data to store, processand transmit for video/audio files can reach into the trillions ofbytes. Mathematical functions used in many lossy encoders/decodersrequire the use of graphic accelerators, faster data processors, largeramounts of memory and higher bandwidth, which is not always feasible,especially for mobile and embedded devices. Another problem with lossycompression is that it cannot be used for all types of data. Every byteof an executable must be precise and without loss, otherwise, theintended meaning, which are the machine instructions, cannot beaccurately processed by the data processor.

There are a wide variety of techniques in the known art that use visualaids to identify data patterns. For example, a time series chart plotsdata values within a two dimensional graph, which enables human beingsto see the structure of the time series over a period of time. The goalof the time series chart is to identify patterns in a visual way. Lines,curves and moving averages may be used to plot the data points in thetime series within the graph. Models are then fitted to the data pointsto help determine various patterns.

A problem using charts to model patterns is that they only tend to usetwo or three dimensions, for they are used for the aid of a human being.They are typically not used for determining the probabilities ofsequences and other characteristics in more abstract spaces, such astopological spaces, non-Euclidian spaces, or in spaces with four or moredimensions. Much of the data processing methods in the known art stillprocess data as a sequence of variables, not as a shape.

Models are also used in computer simulation to generate data. It is nottrivial for a data processing machine to simulate true randomness,though it can generate pseudo-randomness, which is a simulation ofrandomness using techniques such as a stochastic or Markov process. Thisis a problem for encryption, where random data is used to help encrypt adigital message for security purposes. To solve this problem ofsimulating random data by a data processor, known methods have usednatural phenomena, such as weather storms or radioactive decay to assistwith generating randomness. The problem with using natural phenomena togenerate random data is that a data processing system is required tohave additional machines that incorporate data from the naturalphenomena, which may not always be available or practical.

One of the biggest problems in the known art regarding data processingis the theoretical limit of data compression Shannon's entropy statesthe more unpredictable an outcome is, the more data required to describeit. Information theory treats symbols of a message as independent andidentically distributed random variables. This means that one symbolcannot affect the probability of another, for they are independent;unconnected. For example, in information theory, the probability of afair coin landing on heads or tails is considered to be 0.5 and remainsconstant for each flip, no matter how many flips are made. UsingShannon's entropy, it is considered to be impossible to encode, onaverage, the outcome of a fair coin better than 1 bit. Compressionmethods are incapable of compressing random-like data because itsaverage entropy is at maximum. Therefore, when redundant data iseliminated from a message, the probability distribution associated withthe variable usually turns to a normal distribution, for all possibleoutcomes are considered equally probable. A theoretical limit ofcompression and computation of data is generally accepted in the knownart.

In fact, it is not possible to compress all possible files or statesusing a single statistical model, as stated by the Pigeon holeprinciple, for there cannot exist an injective function that can take alarge finite set to a smaller set. In other words, four pigeons cannotfly through three holes at the same time. When variables are consideredto be mutually independent and all their possible states are treated asequally likely, then all possible sequences comprising mutuallyindependent variables are also equally likely. This relates to theaccepted idea that random data is a condition when all possiblesequences are equally probable and is unable to be compressed withoutloss. Data compression methods in the known art are left at an impasse.This is the case for all high entropy data, such as: a binaryexecutable, data simulation, compressed data, encrypted data or simplyrandom data based on a source of natural phenomena, such as radioactivedecay.

U.S. Pat. No. 6,411,228 to Malik titled “Apparatus and method forcompressing pseudo-random data using distribution approximations” filedSep. 21, 2000, however, describes a method and apparatus that claims itcan compresses pseudo-random data by implementing a stochasticdistribution model. In its claims, the stochastic distribution model iscompared with the pseudo-random data. The difference data between thetwo is claimed to be generally less random than the other stochasticdistribution and the pseudo-random data. It is claimed in the patentthat the difference data can therefore be compressed. The differencedata is included with the values required to generate the stochasticdistribution, such as a seed value, which together allow a decoder togenerate the original pseudo-random file.

The problem with this method is that the process that compares thestochastic model to the pseudo-random data is computationally expensive,for the process must compare a large number of stochastic models inorder to find a “best” fit, which is a selection process that leads tothe method generating the difference data. Another hurdle usingstochastic models for encoding pseudo-random data is that the number ofbits needed to describe the seed value to generate the pseudo-randomdata may also be as high as the number of bits required to describe thepseudo-random data itself. In addition, the stochastic models may notalways match random data well enough, for it is generated by computersimulation and not from natural phenomena.

What is needed is a method, article comprising machine instructions andapparatus that can efficiently model the statistics of large sequencesof data, analyze their patterns from a broad view, determine theirprobabilities, eliminate their redundancy and reduce the average entropywithout loss. Because statistical models are the starting point for mostdata processing techniques, any way that allows a data processor toreduce the overall complexity of said models would result in an increasein speed and accuracy of data processing, such as for patternrecognition of human language, pictures and sounds. It would also allowfor the transmission and storage of more data in less time, bandwidth,and space, as well as allow for the determination of a probability valuefor a sequence at a future time using forecasting, and for efficientdata generation, such as random data without requiring devices to readnaturally chaotic phenomena found in nature.

Unless the context dictates the contrary, all ranges set forth hereinshould be interpreted as being inclusive of their endpoints andopen-ended ranges should be interpreted to include commerciallypractical values. Similarly, all lists of values should be considered asinclusive of intermediate values unless the context indicates thecontrary.

SUMMARY OF THE INVENTION

The method, article comprising machine instructions, and apparatuspresented in this inventive material solves many of the problemsassociated with data processing in the known art by identifying a subsetof states out of a superset comprising all possible states of a datasystem. A data system, as presented in this inventive subject matter, isa system where data units are treated similarly to a system of atoms,that being an aggregate of interconnected and mutually dependent unitsthat have non-uniform probability distributions and manifest, as awhole, one state.

Identifiers are created to represent at least one of the following: astate within the state subset and a boundary of the state subset. Theidentifiers allow a data processing device to generate a data systemcorresponding to one of the states comprised by the state subset. Dataprocessing is made more efficient when a number of states comprised bythe state subset is less than a number of states comprised by the statesuperset. When only the states within the state subset are considered,redundancy of information can be reduced, as well as the averageinformation entropy. The result is a method, article comprising machineinstructions, and apparatus that can efficiently model the probabilitiesof a data system's systemic characteristics, such as a large datasequence from the data system, a system's structures and the states ofthe data system itself without incurring a high cost to the dataprocessor and memory as compared to other methods in the known art.

A state subset is identified using a model that describes a datasystem's systemic characteristics; its structures and relationshipsbetween data units. Such a model can be created using a set with addedstructure, such as a space, whereby elements of the space represent apossible state of the systemic characteristic. Probabilities associatedwith the elements are represented by a probability field within thespace, allowing the method, article comprising machine instructions, andapparatus to eliminate redundant spatial forms and structured patterns.

To create a spatial statistical model, the method, article comprisingmachine instructions, and apparatus can receive and process an orderedsequence of units from the data system using a sequence function thatoutputs an output sequence corresponding to the ordered sequence. Themethod, article comprising machine instructions, and apparatus thencorresponds the element of the space to the state of the member of theoutput sequence. The probability of the systemic characteristic, such asthe state of the member of the output sequence, is determined by anumber of correspondences between the element and the member. A spatialpattern is determined based on the structures added to the set, such asa dimension of the space, and the way in which the sequence functionprocesses the ordered sequence.

When the probabilities of the possible states are determined, a positiveprobability subspace is identified within the space, which representsthe data system's state subset. Any state representation of the datasystem, such as an element, comprised by the positive probabilitysubspace indicates a correspondence between the state and the outputsequence is probable. Any element representing a possible state that hasa probability of zero is comprised in another region called a zeroprobability subspace. The system then constructs an ordered list ofidentifiers to represent states within the state subset, or the statesof the members of the output sequences that correspond to the elementswithin the positive probability subspace. By receiving the ordered listof identifiers, a device can generate a sequence or data system in astate within the state subset.

Advantages

There are a number of advantages to the present method, articlecomprising machine instructions and apparatus. The first advantage isthat less memory and storage is required to model the probable states ofsystemic characteristics compared to methods and apparatuses of theknown art, particularly as the number of states rises exponentially dueto an increase in the number of bits representing a word or string.

Other benefits of the present method, article comprising machineinstructions, and apparatus include modeling characteristics of datasystems using polygonal, parametric and continuous functions. Wheremathematical functions in the known art approximate a signal within adomain, the present method, article comprising machine instructions, andapparatus utilizes a variety of functions to model the probable forms ofthe states of the system, which can be used to calculate the probabilityof a particular state deterministically. Polygonal, parametric andcontinuous functions can determine the probabilities of the possiblestates of the data system by selecting elements in the structured set asconstraints, which signify the state subset's boundary. A continuousprobability distribution function can determine probabilitydistributions for the possible states using a continuous probabilitygradient. If required, a discrete probability distribution and aprobability mapping function may also assign probability valuesexplicitly to any state representation utilizing a reference.

For pattern recognition, the method or apparatus can use the spatialstatistical model for matching characteristic patterns of data withknown data systems or probability distributions.

The method, article comprising machine instructions, and apparatus canalso apply the encoding steps iteratively. A sequence function reads theordered list of identifiers as a new data system for the input in asuccessive iterative transformation cycle. A tree data structure canassist with the processing of these iterative cycles. When blocks ofdata are appended in the iterative process, the tree data structureallows the method, article comprising machine instructions, andapparatus to decode only selected blocks or files without decoding allthe appended data blocks.

Another advantage is the ability to create a supercorrespondence of thestates or members of the output sequences, similar to a superposition,where the method or apparatus corresponds at least two states of asystemic characteristic to the same element. Supercorrespondence canreduce the amount of space needed to represent all the possible statesof a data system.

Other modeling techniques include using a structured set of two or moredimensions, using a plurality of structured sets, using spaces with ahierarchical multi-resolution structure, using vectors to plot outintervals of time, and using sets of probability distribution functionsand maps.

Of the greatest advantages from this method, article comprising machineinstructions, and apparatus is the ability to create a spatialstatistical model for random-like data, which allows for itsencoding/decoding, pattern recognition and generation, something whichhas been accepted by various techniques in the known art to beimpossible. Such features include the ability to apply a random-liketransformation function to transform a data system into the random-likedata state when iteratively encoding data. The present method, articlecomprising machine instructions, and apparatus can encode forrandom-like data due to a hypothesis presented in this inventivematerial resting on an idea that random-like data is not a conditionwhere all possible states of a data system are equally probable. Rather,the hypothesis states that random-like data is but one of the possiblestate subset types within a state superset.

Glossary

The term “data system” is used herein to mean a set, aggregate,collection or sequence of mutually dependent units of data, such assymbols, where each has non-uniform probability distributions for theunit's possible states. Such a data system comprises at least two ormore mutually dependent variables or symbols, such as bits. For example,a sequence of 16 bytes may be a data system when the 16 bytes arereceived as and representative of a whole system. Therefore, anythingsimilar to the following may be considered a data system: a computerfile comprising bytes, an encoded message comprising symbols, a documentcomprising graphemes or alphanumeric symbols, a digital picturecomprising pixels, a waveform file comprising samples, a video filecomprising picture frames, an executable comprising machine codesymbols, and a set of data systems in an operating system. Data systemscan remain relatively static within a storage medium. The state of adata system can change due to the units themselves being rearranged andaltered by a data processor and, in a possibly undesirable manner, byphysical forces such as heat that influence a data processor. The stateof a data system can also be influenced by the laws and processes of theunits themselves. The state of a data system can also be random,containing no discernible order, which may be the result of dynamics orforces from the environment, model or a data processing machine, whichmay be unobservable due to limitations of a measuring device. The unitsmay be represented in digital or analog form. In the context of thisinventive material, a data system is not the data processing machine orsystem, which comprises interoperable components that process datasignals, as in a computing device. A data system is not a collection ofindependent files where information does not connect the units together,such as a database or filing system on a storage medium. If, however,the files are treated as one stream or file, and a data processingmachine is required to process the entire set of data in order to accessone file, then the entire collection of data would be considered as adata system. While a data system can be broken into many separate units,such as files, like pieces in a puzzle, the information observed withineach unit are not independent.

The term “data unit” or “unit of data” is used herein to mean componentsor elements of a data system. These units can be; bits, bytes, integers,floats, characters, pixels, wave samples, wavelets, vectors, functions,processes, subsystems and any other units that together contribute to awhole system of data that is intended to communicate a message orrepresent a system's larger state.

The term “information distortion by observation distortion” or“i.d.o.d.” is used herein to mean a measure of the difference between anobserved number of possible states for a data unit and an observednumber of possible states for a system of data units (a data system) asit relates to a value.

The term “mutually dependent variables” is used herein to mean amutually dependent relationship between two or more random variables.For example, a method or system generating one outcome may have aneffect on another outcome. Unlike a Markov process, where theprobability distribution of the present variable affects only theprobability distribution of a subsequent variable, the probabilitydistribution of one dependent variable can affect the probabilitydistributions of all other dependent variables within the sequence orsystem.

The term “space” is used herein to mean a set of elements with one ormore added structures, which gives the elements a relationship thatallows a function to measure certain characteristics between them, suchas a distance between one or more elements. An exemplary space is aphase space, where each point represents a unique state of a system.Another space may be a metric space, such as Cartesian coordinatesystem, Euclidean, Non-Euclidean spaces. A more abstract space would betopological spaces. On the contrary, a sequence comprises members thatcan repeat, where the relationship between members in a sequence isorder only.

The term “member” is used herein to mean a member of a sequence, whichis usually an ordered list of data units, such as the bits in a datastream or the graphemes in a sentence. When dealing within a context ofa space, such members may correspond to elements of a set, though theyare not generally defined as the elements they correspond to. Therefore,an element or point in the space is not a member, for the elements arerelated by at least one structure. Only when a correspondence connectsan element in space to a member can a sequence be endowed withstructure.

The term “supercorrespondence” or “supercorresponding” is used herein tomean corresponding at least two members of at least one sequence to thesame element in a space. It is not unlike the superposition of two ormore items at the same point in space, for corresponding a member to anelement of a space is similar to positioning a member to a point.Technically, the method may not actually place a member on an element,but it can correspond two by a referencing system, mechanism or functionusing a data processing engine and system memory.

The term “state” is used herein to mean a condition, pattern orarrangement of data units. The state can also be an outcome of one ormore variables that occurs at an interval of time or any other measure.For example, each of the 256 patterns of bits in an 8-bit sequence is astate of a data system 8-bit in size.

The term “state superset” is used herein to mean a set comprising allpossible states of a data system.

The term “state subset” is used herein to mean a subset of a datasystem's state superset.

The term “systemic characteristic” is used herein to mean any aspect ofthe data system; a state, structure, unit of data, data subsystem,relationship between units or subsystems, processes, spatial forms, etc.

The term “probability field” is used herein to mean a space where itselements, which represent a possible state of a systemic characteristic,are permitted to be assigned probability values. For example, if thenumber of possible states for a member in a sequence at an interval oftime is 256, then the elements at that interval of the probability fieldwould include states 0-255, where probability values may be determined.For this example, the probability field would not include a pointcorresponding to any value 256 and above because 256 is not one of thepossible states.

The term “spatial redundancy” is used herein to mean information that isnot required to describe the values of the elements of the space.

The term “positive probability subspace” is used herein to mean an areaof the probability field where the points are determined to have aprobability value indicating a correspondence between an element and amember is probable. For example, if a probability value greater than 0indicates an outcome is probable to occur, then a positive probabilitysubspace does not include a point determined to have a probability valueof 0.

The term “zero probability subspace” is used herein to mean an area ofthe probability field where the points are determined to have aprobability value indicating a correspondence between an element and amember is not possible. For example, if a probability of 0 indicatesthat it is impossible for an outcome to occur, then a zero probabilitysubspace does not include a point determined to have a probability valuegreater than 0.

The terms “probability constraint elements” or “probability constraints”or “constraint elements” or “constraint boundary” are used herein tomean the elements, selected within a set with added structure, thatsignify a boundary that contains a positive probability subspace. Forexample, if 5 and 10 where probability constraints, they would “contain”numbers 6 through 9. Another possibility would be that constraints 5 and10 would “contain” numbers 5 through 10.

The terms “probability mapping function” or “probability mapping” areused herein to mean an assignment of probability values toelements/points of a probability field. This can be generated by aprobability distribution function that calculates a plurality ofprobability values based on the probability distribution of at least oneoutput sequence and assigns the probability values to the points withina probability field. It may also be generated discretely for each point.A set of probability values is not a probability map. The values must beassigned to points.

The term “sequence function” is used herein to be defined as a processand a means for receiving a sequence of data units, processing thatsequence using certain argument values and functions and generating anoutput sequence that can be processed back into its original sequenceusing the inverse of the sequence function.

The terms “sequence number” or “state number” are used herein to mean anumber that references a unique sequence or state, usually within a setof states, such as a state subset. This number can be represented by anordered list of identifiers.

The term “base symbol” is used herein to mean the symbol based on thebase number for a logarithm. The base symbol is also a data unit. Forexample, for the equation log₂ 16; the base symbol is 2, which isrepresented by one bit, a data unit. Four bits can describe 16 possiblesubstrates.

The term “transformation cycle” is used herein to mean a series ofprocesses and a means for, by a data processing engine and device, totransform and encode a data system.

The term “generation cycle” is used herein to mean a series of processesand a means for, by a data processing engine and device, to generate anddecode a data system based on the transformation cycle.

The term “pseudo-random data” or “random-like data” is used herein tomean a state of a data system that has no discernible patterns amongstits units of data, and where all possible states of a data unit occurwith relative equal frequency across the data system. It may also bedefined as a state where the average information entropy for each basesymbol of a specified number of bits in a sequence is maximal, based onShannon's theory of information.

Further objects, features, aspects and advantages of the inventivesubject matter will become more apparent from a consideration of thedrawings and detailed description three example embodiments.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a chart displaying operations of the present method, set ofmachine instructions and apparatus;

FIG. 2 is a chart displaying operations of the present method, set ofmachine instructions and apparatus;

FIG. 3 is a chart displaying operations of the present method, set ofmachine instructions and apparatus;

FIG. 4 is an exemplary diagram illustrating a distributed dataprocessing system according to the present inventive subject matter;

FIG. 5 is an exemplary block diagram of a client data processing deviceaccording to the present inventive subject matter;

FIG. 6 is an exemplary graphic chart displaying a distribution ofmembers corresponding to elements of two different spaces, according tothe present inventive subject matter;

FIG. 7 is a flowchart outlining an exemplary operation of constructingan ordered list of identifiers using a positional notation, according tothe present inventive subject matter;

FIG. 8 is an exemplary graphic chart displaying members of an outputsequence corresponding to elements of a set with added structure or aspace, according to the present inventive subject matter;

FIG. 9 is an exemplary of two graphic charts displaying members ofoutput sequences corresponding to elements of a set with added structureor a space, according to the present inventive subject matter;

FIG. 10 is an exemplary of a graphic chart displaying sets of outputsequences, whereby its members correspond to elements of a set withadded structure, according to the present inventive subject matter;

FIG. 11 is an exemplary diagram of the ordered list of identifiers,according to the present inventive subject matter;

FIG. 12 is an exemplary diagram of compressible data blocks, encodingand non-encoding regions, including buffer bits, according to thepresent inventive subject matter;

FIG. 13 is an exemplary diagram of nodes and paths of a tree datastructure for iterative transformation cycles, according to the presentinventive subject matter;

FIG. 14 is an exemplary diagram of nodes and paths of a tree datastructure for iterative generation cycles using a pointer, according tothe present inventive subject matter;

FIG. 15 is an exemplary graphic chart displaying a distribution ofmembers within a set with added structure or a space using a probabilitydistribution gradient function, according to the present inventivesubject matter;

FIG. 16 is an exemplary diagram representing members of an outputsequence being encoded using variable-length codes and a probabilitydistribution gradient function, according to the present inventivesubject matter;

FIG. 17 is an exemplary diagram representing members of an output beingencoded using variable-length codes and multiple probabilitydistribution gradient functions, according to the present inventivesubject matter;

DETAILED DESCRIPTION

The following is a detailed description of the inventive subject matter,which may be implemented by a machine device containing a data processorthat executes a set of machine instructions stored within a storagemedium, such as a hard drive or memory mechanism. The machineinstructions may then be adapted to direct the device to execute themethods described within this inventive subject matter. The presentinventive material may also be implemented by a device or machine thatis made to perform the process and steps without machine instructions,such as a specialized processor or machine. Likewise, the inventivesubject matter may be considered as a process, which can be implementedby a data processor stored in a data processing device.

Overview

The steps and elements of the inventive subject matter are exemplifiedin FIG. 1, FIG. 2 and FIG. 3. Step 101, which is providing access to adata processor. Step 102 is a method and means for receiving, by thedata processor, a data system, which includes a sequence of data units,such as bits, bytes, words, etc. The data system may be stored in astorage medium, which must be accessed by the data processor. The datasystem may also be received through a transmission from an internetconnection and then stored on the storage medium, which may then beaccessed by the data processor. Step 103 is a method and means fordetermining, by the data processor, a state subset of the data system,the details of which are described in the example embodiments. Step 104includes constructing, by the data processor, an ordered list ofidentifiers representing at least one of the following: a boundary ofthe state subset and a state comprised by the state subset. Theidentifiers may be constructed by writing bit patterns unto the storagemedium. Step 105 is method and means for configuring a device togenerate a generated data system that corresponds to the state comprisedby the state subset by providing the ordered list of identifiers to thedevice, which may be allowed by an electronic interconnect from theprocessor to the device.

Steps 106-119 are sub-elements to 101-105. These include: step 106,which is method and means for determining, by the data processor, aprobability of a systemic characteristic of the data system. Step 107 ismethod and means for constructing, by the data processor, an argumentvalue enabling the device to determine the probability of the systemiccharacteristic. Step 108 is method and means for configuring, by thedata processor, the device to determine the probability of the systemiccharacteristic by providing the argument value to the device. Step 109is receiving, by the data processor, the argument value. Step 110 ismethod and means for employing, by the data processor, the argumentvalue to determine the probability of the systemic characteristic. Step111 is method and means for determining, by the data processor, thestate subset based on the probability for the systemic characteristic ofthe data system.

Elements 112-119 derive from steps 101-105. Element 112 states that thestate comprised by the state subset may correspond to a random-likestate. Element 113 is a method and means for employing, by the dataprocessor, a random-like transformation function to the data system,whereby an initial state of the data system is transformed into arandom-like data system corresponding to the random-like state. Elements114, 115 and 116 are comprised by step 104. Element 114 is a method andmeans for determining, by the data processor, a number of statescomprised by the state subset, particularly by an enumerationcalculation. Element 115 is a method and means for accessing, by thedata processor, a generated set of identifiers representing the statecomprised by the state subset. Element 116 is a method and means foridentifying, by the data processor, the state comprised by the statesubset by employing a positional notation. The device can comprise thedata processor, as shown by element 117. Element 118 is a method andmeans for matching, by the data processor, the probability of thesystemic characteristic of the data system to systemic characteristic ofa known data system to recognize a current state of the data system asbeing similar to a state of the known data system. Element 119 furtherexemplifies what a systemic characteristic can be: a state of the datasystem, a state of a unit from the data system, a state of a sequence ofunits from the data system, a structure of the data system, arelationship between at least two systemic characteristics of the datasystem.

FIG. 2 shows steps 201, 203 and 202, which are further comprised by step106 of determining the probability of the systemic characteristic. Step201 is a method and means for initiating a sequence function thatoperated on an ordered sequence of units from the data system togenerate an output sequence that corresponds to the ordered sequence.Step 203 is a method and means for corresponding, by the data processor,the output sequence to an element comprised by a space whereby theelement represents a state of the systemic characteristic. Step 202,which is comprised by step 107, is a method and means for determining,by the data processor, the argument value, particularly by a functionselected from the group consisting of: a polygonal function, a statefunction, a parametric function, a continuous probability distributionfunction, a discrete probability distribution function and a probabilitymapping function that performs at least one of the following: element203, which is a method and means for determining, by the data processor,a number of times the output sequence corresponds to the element;element 204, which is a method and means for determining, by the dataprocessor, a probability of the output sequence corresponding to theelement; element 205, which is a method and means for selecting, by thedata processor, the element as a constraint element that represents theboundary of the state subset; element 206, which is a method and meansfor determining, by the data processor, a probability distribution thatindicates the probability of the systemic characteristic; and element207, which is a method and means for determining, by the data processor,a reference that allows for an assignment of the probability of thesystemic characteristic.

Elements 208-213 further describe elements 201-207. Element 208 statesthat the sequence function is a type of function selected from the groupconsisting of a monotonically increasing function, a summation function,a multiplication function, a trigonometric function, a lexicographicordering function, a sorting function, a sequence function thatgenerates a member of the output sequence that is a measure of avariance from a determined value. Element 209 states that the spacefurther comprises at least two dimensions. Element 210 shows that thespace can be a type of space that is selected from the group consistingof Euclidean, non-Euclidean, topological and phase space. Element 213 ismethod and means for supercorresponding at least two members of theoutput sequence to a mutual element comprised by the space. Element 213is method and means for matching, by the data processor, acharacteristic pattern of the probability distribution to acharacteristic pattern of a known probability distribution to recognizethe probability distribution as being similar to the known probabilitydistribution.

FIG. 3 displays the elements for generating a generated data system thatbegins with step 301, which is a method and means for receiving, by thedata processor, the ordered list of identifiers. Step 302 is a methodand means for generating, by the data processor, the generated datasystem based, at least in part, on the ordered list of identifiers.Steps 306-308 are comprised by steps 301-302. Step 306 is a method andmeans for receiving, by the data processor, the argument value. Step 307is a method and means for determining, by the data processor, theprobability of the systemic characteristic by employing the argumentvalue. Step 308 is a method and means for generating, by the dataprocessor, a generated sequence of units from the generated data systembased, at least in part, on the probability of the systemiccharacteristic. Step 309 is a method and means for initiating, by thedata processor, an inverse of the sequence function to the generatedsequence.

Element 310, comprised by step 307, is a method and means for passing,by the data processor, the argument value to the function selected fromthe group consisting of the polygonal function, the parametric function,the state function, the continuous probability distribution function,the discrete probability distribution function and the probabilitymapping function that performs at least one of the following: element311, which is a method and means for determining, by the data processor,the probability of the output sequence corresponding to the elementbased, at least in part, the argument value; element 312, which is amethod and means for selecting, by the data processor, the elementrepresenting the state of the systemic characteristic as the boundary ofthe state subset; element 313, which is a method and means fordetermining, by the data processor, the probability distribution thatindicates the probability of the output sequence corresponding to theelement; and element 314, which is a method and means for assigning, bythe data processor, the probability of the output sequence correspondingto the element by employing the reference.

Examples of the aforementioned steps and elements of the present method,machine instructions and apparatus are described in greater detail inthe following paragraphs. The following detailed descriptions andexample embodiments should in no way be construed to limit the nature ofthe inventive subject matter. The descriptions and examples providedshould be interpreted to their broadest possible extent.

Description of the Device

With reference now to FIG. 4, which is a representation of a distributeddata processing system, which is a network of computing devices in whichthe present inventive subject matter may be implemented. Distributeddata processing system 400 contains network 407, which is an internetmeans used to provide communications links between various devices andcomputers connected within distributed data processing system 400.Network 407 may include permanent connection, such as wire or fiberoptic cables, or temporary connection made through land-line or wirelessconnections.

In the depicted example, server 401 is connected to network 407, alongwith storage unit 406. In addition, clients 402, 403, 404 and 405 arealso connected to network 407 by connections, such as connection 408.These clients, 402, 403, 404 and 405 may be, for example, personalcomputers or network computers. For purposes of this application, anetwork computer is any computer coupled to a network which receives aprogram or other application from another computer coupled to thenetwork. In the depicted example, server 401 provides data orinformation, such as boot files, operating system images andapplications to clients 402-405. Clients 402, 403, 404 and 405 areclients to server 407. Distributed data processing system 400 mayinclude additional servers, clients, and other devices not shown.

In the depicted example, distributed data processing system 400 is theInternet, with network 407 representing a worldwide collection ofnetworks and gateways that use the TCP/IP suite of protocols tocommunicate with one another. At the heart of the Internet is a backboneof high-speed data communication lines between major nodes or hostcomputing devices consisting of many commercial, government, education,and other computer systems that route data and messages. Of course,distributed data processing system 400 also may be implemented as anumber of different types of networks such as, for example, an intranetor local area network. The connection 408 may use electricity by acable, use light by fiber optics, and electromagnetic by a wirelessmeans. FIG. 4 is intended as an example and not as an architecturallimitation for the processes of the present inventive material.

With reference now to FIG. 5, which illustrates a block diagram of adata processing system that may perform the data processing techniquesdescribed in this present inventive material. Data processing system 500is an example of a type of client computing device, such as a dataserver, personal computer, mobile phone, tablet or other type of dataprocessing system. Data processing system 500 employs a peripheralcomponent interconnect (PCI) local bus architecture 507. Although thedepicted example employs a PCI bus, other bus architectures, such asISA, may be used.

Data processor 501, main memory 505 and enumeration engine 506 areconnected to PCI local bus 507 through PCI Bridge 504. PCI Bridge 504may also include an integrated memory controller and cache memory forprocessor 501. Additional connection to PCI local bus 507 may be madethrough direct component interconnection or through add-in boards. Inthe depicted example, local area network (LAN) adapter 508, Host BusAdapter 502 and expansion bus interface 509 are connected to PCI localbus 507 by direct component connection.

In contrast, graphics adapter 510, and audio/video adapter (A/V) 511 areconnected to PCI local bus 507 by add-in boards inserted into expansionslots, or as integrated chips in the main motherboard. Expansion businterface 509 provides a connection for a keyboard and mouse adapter516, modem 518, and additional memory 519.

In the depicted example, host bus adapter 502 provides a connection fora storage means: hard disk drive 512, Solid State Drive (SSD) 513,CD/DVD-ROM drive 515, and digital video disc drive (DVD) 514.

An operating system runs on data processor 501 and is used to coordinateand provide control of various components within data processing system500 in FIG. 5. The operating system may be a commercially availableoperating system, such as GNU Linux.

An object oriented programming system, such as Java, may run inconjunction with the operating system, providing calls to the operatingsystem from Java programs or applications executing on data processingsystem 500. Machine instructions for the data processing system, theoperating system and applications or programs may be located on astorage device, such as hard disk drive 512, SSD 513 or a DVD-ROM diskusing DVD-ROM drive 515 and may be loaded into main memory 505 forexecution by data processor 501.

Those of ordinary skill in the art will appreciate that the hardware inFIG. 5 may vary widely depending on the implementation. For example,other peripheral devices, such as optical disk drives and the like, maybe used in addition to or in place of the hardware depicted in FIG. 5.Other examples, such as home entertainment devices (gaming consoles, DVDplayers, cable receivers) mobile devices (cellular phones, tablets,portable game devices, cars, etc.) may employ a variety ofconfigurations, alterations, or deletions of said items orimplementations in FIGS. 4 and 5. The depicted example is not meant toimply architectural limitations with respect to the present inventivesubject matter. For example, the method, article comprising machineinstructions, and apparatus of the present inventive material may beapplied to multiprocessor systems. The signals that may be employed bythe data processing system 500, the data processor 501, main memory 505and interconnects 503, 507, 517 used to process data may be electronicusing electrons and other elementary particles, photonic using photons,sonic using phonons and other energetic means such as inducing magneticfields to represent bits of data. These signals may be represented ineither digital or analog form. The data processing device may also usesignals that allow for a superposition of more than one state, such as aqubit of a quantum computer where the qubit superpositions a verticaland horizontal polarization, representing both bits of 1 and 0simultaneously.

The present method, article comprising machine instructions, andapparatus provides a way by which a data system may be processed using adata processing device. The present inventive subject matter may beimplemented on a server device, client device, stand-alone computingdevice, and the like. The present inventive subject matter will bedescribed in terms of functional devices which may be implemented inhardware and software, using the server device and/or client device inFIGS. 4-5, for example. The method described may be embedded inhardware, within a specialized processing unit, such as enumerationengine 506 that calculates enumerations functions faster than thegeneral data processor 501, or from the article and storage means thatcomprises the machine-readable instructions, which may then be stored inmemory or a storage device, such as disk drive 512. The functionaldevices may incorporate one or more elements of the client/serverdevices shown in FIGS. 4-5.

In order to provide the mechanism set forth above, a server, clientdevice, or stand-alone computing device, may be used to implement themethod, article comprising machine instructions, and apparatus describedwithin the present inventive material.

Theory of Operation

The principle of this inventive subject matter rests on the idea thatvariables are mutually dependent and have non-uniform probabilitydistributions as they are representative of a state of a system as awhole, which can be affected by systemic forces and various othercharacteristics. The probabilities associated with states of a unit ofdata can be analogous to the probabilities associated with states ofmomentum of an atom or molecule. A number of states exist in a system ofmatter, which can be divided into four basic phases: solid, liquid, gasand plasma. These phases may be considered subsets within a largersuperset comprising all possible states of the system of atoms andmolecules. Each phase or subset contains states with similarcharacteristics, such as the momentum of the particles. For example,when molecules of water reach higher levels of energy, they shift fromthe solid phase to the liquid phase. Additionally, the molecular shapeand electric charge of the water molecules affects the system's statebased on environmental conditions. The present inventive material treatsa data system in a similar fashion.

Information theory states that random variables from a sequence areconsidered mutually independent and identically distributed, meaningtheir probability distributions are uniform across each variable. Thehypothesis being presented in this inventive material considers randomvariables to be mutually independent only when each variable existswithin a closed system. If the variables reside within a unified system,then the present hypothesis considers the variables to be mutuallydependent and the probability distributions for each variable areconsidered non-uniform. For example, when fair coins are flipped in asequence by the same flipper, they are not, based on the presenthypothesis, considered mutually independent when the coins are beingflipped by the same flipper, which is considered to be one unifiedsystem. This is valid even when the intervals between the flips vary. Arandom pattern of coins is hypothesized to be the result of varioussystemic forces and attributes, which may either be known, unknown orimmeasurable to the observer. The cumulative effects of such forces,however, are not inconsequential, for a measurable effect may beobserved in the course of time.

This idea of mutually dependent variables of a data system relates toanother aspect presented within this inventive subject matter calledinformation distortion by observation distortion (i.d.o.d.), which is ameasure of a difference between a number of possible states for a unitof data, and a number of possible states for the system of units as itrelates to a value, such as a probability distribution. Informationdistortion by observation is hypothesized to occur when a system ismeasured, resulting in units of measurement, which distorts theobservation and thus distorts the information being offered by the unitsof measure. An example would be a probability distribution assigned tothe possible states of the byte, which indicates the probability of eachpossible state the byte can have along a sequence. Because theprobability distribution assigns probabilities to the states of the byteand not to the state of the sequence as a whole, the information beingdescribed by the probability distribution is distorted based on themeasurement of a byte. Any probability distribution assigned to thestates of the byte based on a probable frequency cannot account for allstates of the sequence. For instance, if the probability distributionassigns relatively equal probabilities, where each state of the byte isexpected to occur a relative equal number of times throughout thesequence, then the distribution would not account for all possiblestates of the sequence that do not have all states of the byte occurringwith relatively equal frequency. A probability distribution with littleinformation distortion by observation would assign probabilities to theentire set of possible states of the sequence or data system, where aprobability of 1 (certainty) would be assigned to one possible state,leaving all others a probability of 0 (not possible). A set ofprobability distributions that changes along a sequence of data unitsmay also lessen information distortion, as the probabilities fill theentire sequence. The problem is that any probability model with littleinformation distortion may also be the most impractical using methods inthe known art, such as an indexes, dictionaries, and mathematicalfunctions, for it requires the same amount of information or more todescribe the model than it does to describe the state of the datasystem.

As presented in this inventive material, a measure of the i.d.o.d. canbe utilized to calculate and identify a state subset based on theprobabilities of a data system's characteristics, such as a system'soverall structure in a space. One example would be to calculate all theprobable states of a data system where the possible states for a byteare observed with relative equal frequency. The set of probable stateswould then be determined as a state subset of the data system.

Another way the present method, article comprising machine instructions,and apparatus determines a state subset of a data system is by modelingthe data system's changing probability distributions along intervalsusing a space, which represents a probability field. Probabilities ofthe systemic characteristics are determined by elements of theprobability field, which represent a possible state of the systemiccharacteristic. As a whole, the probability field represents theprobabilities of a system's form, which allows for efficient modeling ofdata sequences of relatively large lengths without incurring a huge costto memory and processing resources, as is the case with methods in theknown art.

An appropriate analogy for understanding i.d.o.d. and the nature of aprobability being assigned to a field or medium in space would be aspinning propeller of an airplane. The propeller may have only twoblades. Yet as they spin, the blades appear as a blurred disk due to theeye's persistence of vision. At that moment, the exact position of theblades cannot be measured by the eye. However, the observer is viewingthe blades from a much broader perspective, where all the possiblepositions or states of the two blades can be seen in one moment whentheir blurred motion takes on a shape. If the observer were to take apicture of the spinning propellers using a high-speed shutter camera,the position of the blades would be known at that exact moment in time,yet their velocity would not. Thus, a distortion is brought to bear. Theobservation has been distorted to a single frame, and therefore, so hasthe information about the spinning blades, which has now become a morefinite observation than when the blades are seen in motion. If one wereto treat a sequence of pictures made from a high speed shutter camera atvarious intervals as mutually independent variables with norelationships between them, then it would be impossible for an observerto know the true nature of the propellers, which is that they arespinning.

An analogy of a data system's structure and i.d.o.d may also be seenwith motion pictures. A movie appears to unfold before the eyes of theobserver. And yet, the movie has already been made; the rolling sheet ofcelluloid travels across a light in a projector. As the rapid successionof frames creates an illusion of movement, the interconnectedness of theframes can be observed, the structure of the movie, which is lies beyondthe single fame. While a moment to an audience may be experienced as onesecond, that moment contains 24-30 picture frames. Looking further in,the frame comprises other moments, which are the millions of pigments inthe celluloid (or digital pixels). Moving further, the pixels arerepresented by a series of bytes. And finally, the bytes make up asequence of bits. The definition of the outcome changes by the way themovie is measured. Measuring a system results in a distortion of theobserver's view from the system's higher structure, which in the analogyof the motion picture, is the characters, the story, the movement . . .the movie's systemic characteristics.

Another feature to modeling probabilities of systemic characteristicsusing a space is that a compression of the probability field may beachieved using a sequence function that structures a state dimension sothat the members of one sequence supercorrespond with members of thesame sequence or of different sequences. For example, when the number ofstates of a data unit is summed monotonically along a dimension of time,many of the members of all the possible sequences correspond to mutualelements. When the sequences supercorrespond in this way, lessinformation is required to describe all the possible states representedby the space, for the field is essentially compressed.

For example, a byte can represent 256 possible sequences of bits.Assigning probabilities to each state result in 256 assignments.However, if a space represented the 256 possible states as monotonic8-bit sequences, where the x axis signifies the length of the sequencein bits (the time dimension) and the y axis signifies a summation of thepossible states of the bit (1 or 0) at each interval on x (8 being thehighest value on y), then the number of possible elements to associateprobability values to in this space would be 44, not 256, since many ofthe possible sequences correspond to mutual elements. Likewise, for adata system of 128 bits, the number of possible states is 3.4028×10³⁸.However, the number of elements in a space containing all the possiblemonotonic 16 byte sequences, where the numeric value of the state ateach interval of the sequence is added to the previous member, is34,696, which is a significantly lesser number to determineprobabilities for, yet it accounts for all the possible states of the128 bit data system.

A set of identifiers that represent each state comprised by the statesubset need not be constructed, either statically or adaptively, orappended with the ordered list of identifiers as is the case for anindex or dictionary in the known art. The identifiers can be constructedusing an enumeration calculation, which determines the probable stateswithin a state subset type, such as a random-like state subset type,which an indicator may denote along with the ordered list ofidentifiers.

In addition to what has been previously described, a variety ofdifferent functions may also be used to assist in determiningprobabilities of systemic characteristics, other than using a functionthat explicitly assigns a probability to a state of the state subset.One such function may select constraint elements that represent a statesubset's boundary. For example, if a one dimensional line contains 100elements and element “30” and “70” are selected as constraints, then thetwo will delineate areas of space to establish a state subset. If thepositive probability subspace is “inside” the boundary, all elementsbetween “30” and “70” will be determined to have a probability value of0.025, as well as the corresponding state the element represents. Usinga referencing calculation, an identifier representing element “31” wouldbe read by the decoder as the first element “0” whereas element “69”would be read as element “39.” The referencing calculation enables thedecoder to know that identifier “0” refers to element “39”, identifier“10” refers to element “49” and identifier “39” refers to element “69.”Any state corresponding to elements “30-70” are considered to be withinthe state subset, whereas all other states represented by elements below“30” or above “70” are considered not probable or outside the statesubset's boundary. Using this technique, the number of probable statesis reduced to 40, where there were originally 100. Such functionalitycannot be achieved without adding structure or relationships to thestates. A variety of boundaries can be created using polygonal andparametric functions that can generate lines, polygons, curves and othersmooth shapes to facilitate the determination of the subset boundary.

Implementing the hypothesis regarding data systems and informationdistortion by observation distortion allows the present method, articlecomprising machine instructions, and apparatus to compress and generaterandom data efficiently. When randomness is considered as a type ofstate subset, and the number of states within the random-type subset isless than the number of all possible states of a data system, then theaverage information entropy of the data system in the random-like statecan be reduced.

FIG. 6 includes a graph 600 that depicts a space comprising a multitudeof members 605 of a series of monotonic sequences from a data system inthe random-like state. The members 605 appear as dots corresponding toeach interval on the x axis 608. The dots in graph 600 appear asvertical lines at each interval, but this is merely an effect due to thedots being in close proximity to each other and because the sequencesare superimposed in the graphs due to supercorresponding the members tomutual elements. The members gather around a mean value of 2040, whichis indicative of a normal distribution. Because the set of randomsequences reveals their systemic characteristic as a pattern within theprobability field (where the corresponding elements do not take up theentire space), the method can use the model to describe the random statesubset type.

Using a structured set or space as a means to model the probable statesof a data system is by no means a limitation of the present inventivematerial. Any method or apparatus that is able to identify the state ofa data system as corresponding to at least one state within a statesubset out of its state superset, applies to this method, articlecomprising machine instructions and apparatus. The following sectionscontain example embodiments.

First Example Embodiment

The present description of a first example embodiment describes how thepresent method, article comprising machine instructions, and apparatuscan decrease the average entropy of a random-like sequence of 16 bytes.This is accomplished by determining a random-like state subset based ona spatial model. The description goes on to describe how to encode thesequence, a state within this state subset, by enumerating the number ofprobable sequences and using a positional notation to encode the state.These steps are described in the following stages: first, determining astate subset for random-like data system; second, receiving a 16 bytedata system/sequence and constructing the ordered list of identifiers torepresent the state of the sequence comprised by the state subset;third, configuring a device to generate the 16 byte random-like sequenceby providing the ordered list of identifiers to the device.

Spatial Statistical Model

One of the ways of determining the state subset of a data system is todetermine the probability of its systemic characteristics within thecontext of a set with added structure such as a space. In this firstexample embodiment, a spatial statistical model is constructed forrandom-like data, although any type of data can be modeled using aspace.

The process of creating a spatial model begins with establishing the setwith added structure. In this present embodiment, the space is aEuclidian/phase space with two dimensions; a state dimension signifiedby a y axis, and a time dimension signified by an x axis. The statedimension refers to the possible states of the systemic characteristic,which in this example is the state of a unit of data; one byte. FIG. 6shows two such spaces; space 600 and space 612. The visible points 605and 616 represent the elements that correspond to members (bytes) ofoutput sequences processed by a sequence function. Space 600 shows thecorresponding members of output sequences from a random-like data file.Space 612 depicts the corresponding members of output sequences from abitmap image file. The two spaces are presented to show how differenttypes of data systems may have dissimilar structural patterns in space.It is the creation and utilization of space 600 that is used in thefirst and second examples as the model for a random-like data system.

The structured set or space can be recorded in computer memory as anarray comprising two dimensions. Programming languages, such as C andC++, allow the coding of array or vector data types. An example of howthis may be written would be as follows: int space [16][4080], where“space” is the name of the set or integer array with 16*4080 elements.The structure that is being added is the relationship of the 4080elements that correspond to each of the 16 elements, which gives 16*4080total elements. In this present embodiment, the space, as exemplified by600 and 612, is constructed to have two dimensions; one for x, 608 and619, and one for y, 609 and 620, which creates a 2 dimensional space or,a 2 dimensional array referenced in a data processing system's memory.Because 16 byte sequences are modeled in the first example embodiment,the maximum interval value for the x axis is 15, which is the number ofmembers in the output sequences from 0-15. The y axis signifies thesummation value of the states, which in this example is 0-4080, as themaximum value for an 8 bit symbol is 255 (0-255): 255*16 is 4080. Whenplotted using the sequence function, the members create a monotonicsequence in the context of a two dimensional space. The process of thesequence function is explained in subsequent paragraphs.

Using the space, the current state of the members of an output sequenceis matched to the element of the space. To do this, the process mustknow how a member is defined. In this first example, the process uses asymbol set criteria, which defines what a unit of data is and how thedata units should be read. The criteria for the symbol set may be:

1. A number of bits: For example, the system could choose 4 bits, 8bits, 16 bits or any number below the total size of the data system.

2. A selection of symbols to use out of the total possible based on thenumber of bits determined in #1 above. For example, the system canchoose 10 symbols out of the total 16 possible symbols of a 4 bit set.

3. The locations in the data stream to change the criteria of the firsttwo options. This allows for a dynamic change to the types of symbolsthe method can use. As an example, it may read a 4 bit symbol, then an 8bit symbol, and then a 24 bit symbol, and repeat this cycle throughoutthe data stream.

In the first example embodiment, the symbol set is defined as 8 bits inlength (1 byte), where all possible states of said bit length areallowed until it reaches the end of the data stream. However, asexemplified above, this particular choosing of said criteria of a dataunit is not a limitation of this aspect of the method or apparatus andshould be interpreted as a broad guideline for what is possible.

To model a state subset of a data system using the spatial statisticalmodel, the method must read an adequate number of data systems withsimilar systemic characteristic patterns. The modeling begins withreceiving a number of known random-like data systems by a sequencefunction that reads an ordered sequence of 16 bytes from the datasystem. For example, if a data system were 16,000 bytes in size, thesequence function reads a sequence of 16 bytes, and then reads the 17thbyte as the 1st byte in a new 16 byte sequence. This would split thedata system into 1000 ordered sequences. The sequence function shouldread an adequate number of ordered sequences from as many similar datasystems as possible in order to create an accurate model. The presentexample does not specify the number of data systems to use, for thepossibilities are broad where no limitations need be provided orinterpreted.

It should be made clear that the spatial model of the data system isbased, in part, on how the sequence function processes the orderedsequences. Considering the possibilities, the present method, articlecomprising machine instructions, and apparatus is able to model datasystems in a number of different ways dependent on said criteria. Forexample, the sequence function can model the 16,000 byte data systemwithout dividing it into 16 byte sequences. The sequence function canprocess a single output sequence 16,000 bytes long. The sequencefunction could read sequences 256 bytes long. Splitting the data intosmaller sequences, such as 16 bytes, allows the method to model a moremanageable number of probable states in the spatial statistical model,for the greater the size of the sequences, the greater the number ofpossible states to model. However, splitting the data system intomultiple sequences is not a requirement. Additionally, a sequencefunction need not be limited to reading the ordered sequence from thedata system sequentially. As previously stated, based on the symbol setcriteria, the sequence function may read the data sequence from the endto the beginning, reading rows from right-to-left, then left-to-right.The read pattern may also be a random-access type read, where thesymbols are defined on a 2-dimensional array. There are manypossibilities available to a sequence function, which should beconsidered to its broadest extent. Any ordered sequence based on thesymbol space criteria can be used.

In this present embodiment, the sequence function inputs the orderedsequence, reads each byte sequentially and outputs a monotonic outputsequence, where each state value of all preceding members (bytes) areadded to the state value of the current member. It performs this foreach 16 byte sequence. For example, if the values for the sixteensymbols within a sequence were to appear as follows; 2, 189, 240, 52, 0,19, 25, 123, 207, 10, 10, 81, 147, 94, 202, 248; the resulting outputsequence would be; 2, 191, 431, 483, 483, 502, 527, 650, 857, 867, 877,958, 1105, 1199, 1401, 1649.

From the many possibilities contemplated, the sequence function mayprocess the ordered sequence using a variety of methods beyond thesummation function presented in this first example embodiment. Anysequence can be generated by the sequence function as long as theinverse of the sequence function generates the ordered sequence. Forexample, the sequence function can multiply each successive member inthe ordered sequence; if the members in an ordered sequence are 10, 5,8, and 7, the output sequence would be 10, 50, 400, and 2800, eachmember being multiplied by the previous product. In another embodiment,the sequence function may be a trigonometric function along with asummation function, where the output sequence takes the shape of acircle or spiral within a three dimensional space. For example, whenusing a trigonometric-summation sequence function, the state value ofthe member of the ordered sequence can determine the length of theradius at each step of creating the circle. The value of the radius forone member may then be added to the radius of the preceding members,where the state values of the members at each step are added to thestate value before it. The output sequences will therefore create anexpanding tube or spiral form in space. If the summation function is notapplied when using a trigonometric function, then the tube or spiralwill not expand, but have a straight uniform shape, as a tube orcylinder.

In another embodiment, the sequence function may use two differentfunctions; a summation and a sorting function, which outputs twoseparate output sequences. In that embodiment, the first output sequencewould be the state values of the ordered sequence sorted inlexicographic order and then made monotonic by the summation function.The other would be a sorting sequence, where the state of each memberindicates a sort value that enables the inverse of the sequence functionto arrange the members in the lexicographic order back into theiroriginal unsorted sequence. For example, if the original orderedsequence was the following; 42, 150, 0, 96, 250, then the lexicographicoutput sequence would be 0, 42, 96, 150, and 250. The first member ofthe sorting sequence would have a value of 2, for “0” is the thirdmember of the ordered sequence, and therefore must be moved 2 slots fromthe first slot, whereas 42 and 96 are pushed to slots 1 and 2. Theresulting sequence would be 42, 96, 0, 150, and 250. Slot 3 is marked asset and will not be counted in subsequent iterations. The second memberof the sorting sequence would be “0”, since “42” is now the first memberof the sequence in the current iteration and thus is not required to bemoved. The first slot is now marked as set. The third member of thesorting sequence is 1, for “96” must be moved one slot to be in thefourth slot as it skips over the third slot, which has already beenmarked as set. Because “150” is now in the second slot in the nextiteration, the final member of the sorting sequence is 0. The resultingsorting sequence is then as follows: 2, 0, 1, and 0, which may be thensummed to generate a monotonic sequence; 2, 2, 3, 3.

In another embodiment, the sequence function can output delta values,which represent the difference between two or more members. For example,based on the preceding sequence of 0, 42, 96, 150, and 250, its deltasequence would be 0, 42, 51, 54 and 235, where the members measure thedifference between a current member and a preceding member.

In another embodiment, the sequence function may generate an outputsequence where its members represent a measure of a variance from adetermined value, such as a mean. For example, if the determined valueis 128 and the state value for the member of the ordered sequence is255, then the sequence function would output a value of 127 for thecorresponding member of the output sequence.

Whatever way the sequence function outputs, it establishes, along withthe structure of the space, how the data system is to be modeled todetermine the spatial pattern inherent in the ordered sequence.

The next step is to call a state function that determines acorrespondence between the members of the output sequences and theelements. When a number of correspondences is determined, theprobability of the systemic characteristic can be determined. Matchingthe state of the member to the corresponding element gives structure tothe sequence within the space. Out of the many possibilitiescontemplated, one way the state function can determine thecorrespondence between a member and an element is to add a value of 1 toall the elements of the data structure stored in computer memory eachtime a member corresponds to an element. For example, if the state valueof the first member is 50, then a 1 would be added to the element of thedata structure corresponding to the first element of the time dimension,and the 50th element of the state dimension. For the second member, ifits state value is 250, a 1 would be added to the element referenced bythe second element of the time dimension and the 250th element of thestate dimension. In this manner, the number of times a membercorresponds to an element is summed in the array data structure. Amongthe other possibilities, a third dimension may be employed thatreferences the output sequence. For example, if there are 1,000 outputsequences, the state and time dimensions would correspond to 1000elements of the third dimension where each of the 1,000 output sequencesare represented from 0-999. Using this technique, every output sequencewill be modeled in its own two dimensional space; a value of 1 beingadded to the element corresponding to one of the one thousand outputsequences. These are possible techniques for corresponding members tothe elements of the structured set, and should not be considered tolimit the scope of the present method, set of machine instructions andapparatus, rather they should be perceived as only examples within abroad set of possible examples.

The next step in constructing the spatial statistical model is todetermine the probabilities for the possible states represented by theelements. Out of the many possibilities contemplated, a boundarygenerated by a polygonal, parametric and continuous function can help todetermine the probability values. The boundary delineates between areasof the space containing elements that are likely to correspond to outputsequences and those that are not. This can be done using a “curvefitting” technique, where the output of a polygonal and spline functionconforms to a distribution area comprising elements that correspond tomembers. Constraint elements are selected where the boundary, created bythe polygonal, parametric and continuous function, intersects an elementof the space. Those elements are selected as constraint elements orboundaries of the positive probability subspace, which represents thestate subset. Such polygonal, parametric and continuous functions cangenerate a variety of shapes, such as lines, polygons, parametric shapesand curves. In FIG. 6 in space 600, particularly the probability fieldbound by boundary 611, an output of a continuous function is shown aslower boundary 601 and upper boundary 610, which specify the positiveprobability subspace 602 and zero probability subspaces 603 and 604.Space 612 and its probability field outlined by boundary 613 hasconstraint boundary 615 that defines positive probability subspace 618and zero probability subspace 614. For space 612, the positiveprobability subspace reaches all the way to the x axis and takes morearea of the probability field than positive probability subspace 602 inspace 600.

Out of the many possibilities contemplated, the curve fitting algorithmmay begin with a line segment, where the first endpoint has a value of(256, 0) on the y and x axis respectively. The second endpoint of theline has a coordinate value greater than 4080 on the y axis and 15 onthe x axis. This line segment is an upper boundary constraint that isgiven a y value that places the line segment in a region where nomembers correspond to elements. Iteratively, the process changes theshape of the boundary by lessening the argument values of the linefunction. It then checks if the line hits any element/point thatcorresponds to any member of the output sequences. If it does not, theprocess repeats. If it does, the process stops and the argument valuesfor the function are constructed. This same process is completed for alower boundary, where the coordinates (the argument values) of theendpoints are (0, 0) and (0, 16), y and x axis respectively. Theiterations continue, the argument value increasing after eachintersection check, until the line hits an element/point thatcorresponds to a member. One of ordinary skill in the art of “curvefitting” may use other known methods, such as math functions that fitthe continuous function to the distribution of the correspondingmembers. The aim is to match, as closely as possible, the output of thecontinuous function to the shape of the distribution of the membersbased on their correspondence to the elements. The argument values maybe used by the polygonal, curve and parametric function to determine theprobabilities of the states without going through the steps of the“curve fitting” algorithm.

Inside the positive probability subspace, defined by the constraintelements, each point is determined to have equal probability values. Allpoints outside the constraints have a probability value of 0. While theprobability values can be assigned to every possible state representedby an element, there is no need in this present example, for theconstraints are used in an enumeration calculation that outputs a numberof probable sequences within the positive probability subspace. Theconstraint elements limit the ranges the possible states of a member canbe. The enumeration calculation counts only the output sequences withinthe boundary and thus innately determines the number of sequences thathave a probability greater than 0 by counting only the sequences withinthe positive probability subspace, making any state beyond theconstraint impossible and therefore uncounted. The enumerationcalculation is described in subsequent paragraphs. It should be notedthat points “inside” or “outside” the boundary may be reversed ifdesired, where points outside the boundary are determined to haveprobabilities greater than 0, and points inside the boundary aredetermined to have probabilities of 0.

Once the probability constraints are selected, they can form theconstraints of a model, or they can be compared with other modelspreviously stored in a storage medium. If the constraints fit thoseknown models, the data may be considered to match that model. Forexample, if a set of output sequences are analyzed in a space using theconstraints described above, and those constraints fit inside or matchthe constraints of a previously recorded statistical model forrandom-like data, then the model being tested will be determined as arandom-like data model. Likewise, the present method, article comprisingmachine instructions, and apparatus may test the state of a data systemby determining whether or not its shape fits in the boundaries of apreviously recorded model. Said test is described in the followingsection.

The model for random-like data in this first example specifies theconstraint elements at the 15^(th) interval of the output sequences as,(15, 3,430) and (15,600), x and y respectively. In FIG. 6, space 600shows upper bound constraints 606, and lower bound constraints 607,which are aligned with upper boundary 610 and lower boundary 601. Forspace 612, upper bound constraint 617 has a higher state value of 3500and has no lower bound constraints. For the members at the 15th intervalof space 600, the upper and lower bound probability constraints are atthe following x and y values, (14, 3,260) and (14,500) respectively. Theconstraint elements may continue until x is 0, the first interval. Withthe argument values constructed, the probability values for the possiblestates represented by the elements in the space determined, the spatialstatistical model for random-like data is now established and can bestored in a storage medium for future processing. The probabilities ofthe systemic characteristics represented by the probability field formthe basis of the spatial statistical model, whereby its argument valuescan be passed to the function that aided their construction.

In this example, the spatial model described previously is used to lowerthe average maximal entropy of 128 bits for a random-like data sequenceof 16 bytes. To accomplish this, the system constructs an ordered listof identifiers based, in part, on the state subset of the data system.In the present example, these identifiers describe the members using aunique sequence number that encodes the state value of each member of anoutput sequence as a radix using a positional notation. In this example,the identifiers are based on an enumeration calculation, whichcalculates the total number of output sequences (or states) within thepositive probability subspace (or state subset), which is defined by theconstraint elements described previously. The same positional notationmay then be used by a decoder to generate the 16 byte sequencerepresented by the identifiers.

Matching Data

To lower the average entropy of 16 bytes of random-like data using thespatial statistical model, the data processor first receives the datasystem comprising 16 bytes intended to be encoded. The system initiatesthe same sequence function described previously, which processes theordered sequence (the data system) and outputs the monotonic outputsequence.

The system must then determine the state subset for the data system. Todo this at this stage, a matching test can be performed to determine ifthe output sequence is similar to the systemic characteristics of aknown spatial model for random-like data. A space is constructed, wherethe output sequence corresponds to the elements.

From the many possibilities contemplated, this test may be accomplishedusing an intersection test, where the elements of the spacecorresponding to the output sequence are tested by the data processor todetermine whether or not the output sequence matches or resides withinthe positive probability space of the random-like spatial model. If themembers test positive for matching or residing within the positiveprobability space of the random-like spatial model, then that model isapplied to the output sequence to encode it. If not, the output sequencecan be tested with other spatial statistical models stored in computermemory or storage device, until a match is made.

The test may be efficient due to the fact that only elementscorresponding to the output sequence and within the boundary of themodel would be tested. There may be a binary test, where elements withinthe positive subspace of the spatial model are assigned a value of 1,where all other elements are assigned a value of 0. The structure andnumber of elements of the space corresponding to the output sequence andthe spatial model would be identical. The intersection test can be astraightforward algorithm that reads the elements in the spacecorresponding to the output sequence and tests whether those sameelements in the spatial model have a value of 1. If the elements beingtested correspond with the same elements of the model, the data systemor output sequence being tested will be considered as fitting the model.This test is similar to the methods using collision detection andvoxels, as in U.S. Pat. No. 8,279,227 to Cohen titled “Method fordetecting collisions among large numbers of particles” filed Apr. 28,2008. This form of pattern recognition can be used for a variety ofsystemic characteristics. Such an algorithm given in the present examplemay not be considered as a limitation to matching known systemiccharacteristics of data to a known model or other data.

When a match is made, the constraint elements from the matching spatialmodel are received as argument values by the data processor and passedto an enumeration calculation function.

Ordered List of Identifiers

The next step is to encode the members using a positional notation thatimplements an enumeration calculation, which determines the total numberof probable sequences that obey the model's constraints. While thesystem could iteratively count and test each probable sequence to see ifit obeys said probability constraints, this would be verycomputationally expensive, even though such a technique can be used andtherefore falls within the scope of this method, article comprisingmachine instructions, and apparatus. In the present embodiment, theprinciple of Inclusion-Exclusion (P.I.E. or inclusion-exclusion) isutilized to determine the number of states within the boundary of thestate subset or the positive probability subspace. The formula for theP.I.E. is as follows:

|∪_(i=1) ^(n) A ₁|=Σ_(k=1) ^(n)(−1)^(k+1)(Σ_(1≦i) ₁ _(< . . . <i) _(k≦n)|A _(i) ₁ ∩ . . . ∩A _(i) _(k) |)

A simple example of how to use inclusion-exclusion to enumerate allpossible monotonic sequences that sum to a value of 4080 is described inthe following paragraphs. Algorithmically, the formula may be expressedusing the following steps.

The total number of probable output sequences is calculated using thebinomial coefficient(_(k) ^(n)), where n is a total number of possibleelements to choose from, such as a total value that all 16 members in asequence add up to (i.e. 4,080), while k is the number of selectedelements, which in this case, is the length of the sequence minus one.The number of elements between each of the 15 selected elementsindicates the state value of a member. The algorithm calculates all thepossible ways that n can be divided by k. An example would be thefollowing: n=4095 and k=15, which gives the total number of possibleoutput sequences comprising 16 members that sum to 4,080.

If there were no limits on what the state value could be, such as astate value between 0-4080, then the solution would be the non-negativenumbers to a total, such as: (m1+m1+m3+ . . . m16)=4080. However,because each member can only have a value between 0-255, the processmust discard the solutions with m1≧256, which is m1−256≧0, orm1*=m1−256; where (m1*+m1+m3+ . . . m16)=3824. The number of probablesequences meeting this first condition is calculated using the binomialcoefficient,

$\begin{pmatrix}3839 \\15\end{pmatrix}.$

The first operation subtracts the number

$\quad\begin{pmatrix}3839 \\15\end{pmatrix}$

from

$\quad\begin{pmatrix}4095 \\15\end{pmatrix}$

16 times, once for each member, which results in the total number of allsequences meeting the first condition. Using the principle ofinclusion-exclusion, this first set of calculations in the series ofcalculations over counts the number of probable sequences subtracted,for it included sequences that did not meet all the conditions for eachmember. It must add back the total number of possible sequences wheretwo pairs of members with a value between 0-255 are considered. This iscalculated in the second set of calculations, such as (m1*+m2*+m3+ . . .m16)=3568, where m1 and m2 are paired. The next set of enumerationcalculations in the series subtracts the total number of probablesequences where three pairs of members with values between 0-255 areconsidered. This multiple set of calculations repeats, alternatingbetween sets of addition and subtraction, to and from

$\quad{\begin{pmatrix}4095 \\15\end{pmatrix},}$

each time increasing the number of members considered with valuesbetween 0-255, until a final number results, which is the number of allprobable sequences that sum to 4080.

Taking the inclusion-exclusion technique further, all probable sequencesthat sum from the range of 600-3430 can be determined, which is betweenthe constraints at the 15^(th) interval on the x axis. To do this, theprocessor must perform the enumeration calculation described above forall output sequences that have a sum value between 600-3,430. Forexample, the method begins calculating the number of probable sequenceswith 16 members summing to 3,430. The process stores this number incomputer memory, decrements the total to 3,429, starts a new enumerationcalculation, then adds the resulting number to the final tally. Itcontinues until the last sum is 600, at which point the iteration stops.The result is the total number of probable sequences where their totalsare within the range of 600-3430, as determined by the constraints,which in this example equals 3.402822×10³⁸.

Among the possibilities contemplated, a total number of output sequencesobeying the constraint at the 15^(th) interval can be reduced when theother constraints along the x axis are also used to eliminatenon-probable sequences. Based on the constraint selected at the 14^(th)interval on the x axis, the range is 500-3260. Any sequence that sums to3320 cannot have their 14^(th) member go beyond the y value of 3200.Because those output sequences were included in the enumerationcalculation described above, the method can subtract those sequencesfrom the final number. This is calculated by finding the number ofsequences where the 15th member has a state value within the range of3261-3430 and 0-499, yet where the 16^(th) member has a state value alsowithin range of 500-3260. Using the inclusion-exclusion method, thenumber for these output sequences is: 2.481976×10³¹. This subtractionmay continue for the next constraints at the 14th interval.

To calculate the measure of the entropy for a random sequence of 16bytes, using the spatial probability model as described previously, onecan use the formula: log₂ n, where n is the total number of probablesequences.

Based on this equation, the information entropy of a random-like 16 bytesequence using the statistical model for random-like data, as presentedin this example, is 127.9999996 bits on average. This is less than themaximum entropy of 128 for a 16 byte message without using the model.

To encode the members of the 16-byte sequence, the method or apparatususes the same inclusion-exclusion enumerative calculation to encode aunique sequence number from 1-3.402822×10³⁸ using the positionalnotation. The number is stored as part of the encoded message of theordered list of identifiers using bit fields.

A variety of positional notation systems exist in the known art, such asmixed radix and non-standard positional notations. Arithmetic coding isa popular method in the known art that also uses a numeral system. Outof the many possibilities contemplated, a possible algorithm to encodethe unique sequence number using positional notation is described below.

Each member of the output sequence is represented by a radix of thepositional notation. The positional notation algorithm must match thevalues of each radix to the values of the members of the outputsequence, whereby a final sequence number is encoded, which representsthe state comprised by the state subset. The range of each radix valueis determined by the range of possible states for the members set by theconstraint elements. The process of encoding the sequence number beginswith the first member in the sequence.

As shown in FIG. 7, flowchart 700, because the order of the probablesequences starts with 1, the encoding process must calculate the numberof probable sequences that sum to a value that is above the lowestconstraint or sum, and below the sum of the output sequence, step 702.This number is added to the sequence number, for the current outputsequence is ordered above the other output sequences with a lessor sum.For example, if the sum of the output sequence is 2040, and the lowestsum is 600 at the 15^(th) interval on x, then the number of all probablesequences with a sum from 600 up to 2039 would be determined and addedto the sequence number.

The algorithm now checks whether there are any members following thecurrently evaluated member in the output sequence, step 703. If true,the process continues, step 704. Because the first member generally isfollowed by others, this would be usually true at this stage.

At step 704, the algorithm compares the radix value to the actual statevalue of the currently evaluated member. If the radix value is not equalto the state value of the member, the algorithm proceeds to step 705,where an enumeration calculation commences, which determines the numberof probable output sequences comprising all subsequent members which addup to the sum of the output sequence. The radix value is thenincremented by 1, step 706. The algorithm then decrements a “proxy” sumof the output sequence, steps 707, which will then be used in subsequentenumeration calculations to encode the current radix value. This resultof the enumeration calculation is added to the sequence number, step708, which will ultimately result in becoming the uniquely encodedsequence number.

The next step tests whether the state value for the currently evaluatemember equals this incremented radix value in step 709. If the twovalues are equal, the process moves to the next member in the sequence,step 710, and repeats said steps from step 703, whereby the number ofmembers to calculate for goes down to 14. If the radix value and theactual value for the currently evaluated member do not equal, anotherenumeration calculation begins, step 705, which calculates the number ofprobable output sequences having 15 members that obey the constraintsand proxy sum, and adds the number to the final sequence number, step708. It then continues with step 709, checking if the radix value andthe member equal. This process repeats for all the members of the outputsequence until the last member is reached. The process tests the lastradix value, step 711, and increments this radix, step 712, until itsvalue matches the real member's state value. The ordered list ofidentifiers is constructed to represent the sequence number, step 713.The result is a uniquely encoded sequence number that describes thestate values of each member of the output sequence, step 714, signalingthe end of the algorithm.

For this present first example embodiment, the following may be includedin the list of identifiers:

-   -   1. Symbol Set criteria (if required).    -   2. Size of the data system.    -   3. The argument values for the continuous functions or a 1-bit        signal confirming whether the data matches a predetermined        model, such as pseudo-random/random-like data system.    -   4. The sequence number describing the state comprised by the        state subset (members of the output sequence that are within the        positive probability subspace).

If the symbol set is not known (not included in the decompressor) thenit is required to include that information with the ordered list ofidentifiers. This is usually not required for random-like data, as allthe 8-bit symbols have relatively equal frequency of occurring along thedata system. One of ordinary skill in the art may understand that adecoder may also have default values for certain identifiers and notrequire them to be part of the encoded message. For example, thedefaults for the symbol criteria may be 8 bits, the size of a messagebeing 16 bytes, and the argument values satisfying the constraints forpseudo-random/random data.

Generating a State within the State Subset

The sequence number is decoded by reversing the algorithm of thepositional notation system used to encode the members of the originaloutput sequence. The decoder must receive the ordered list ofidentifiers and the argument values of the model for the random-likestate subset type to generate the data system. In this example, thespatial model uses the constraints determined for a random-like datasystem, which are passed as argument values to the function that selectsthe constraint elements; its output intersecting elements in the spacerepresenting the boundary of the state subset. This is done for thelower and upper boundaries as described previously using a line segment.

The process then calculates the total number of output sequences withinthe positive probability subspace using the enumeration calculation aspreviously described. Calculating this number enables the presentmethod, article comprising machine instructions, and apparatus to readthe correct number of bits required to describe the unique sequencenumber, which is between 0 and the total number of output sequencesobeying the model. Finding the correct number of bits can be solvedusing log₂ n. When the identifiers representing the sequence number arereceived, the decoding of the output sequence may begin.

The next step is to determine the approximate sum of the encoded outputsequence. This may be accomplished by determining the number of probableoutput sequences, which sum within the range established by the model.This number is added to a final tally, where it is compared with thesequence number. The comparison test is done by the data processor todetermine if the sequence number is greater than the final tally. Ifyes, the data processor continues with incrementing the sum to be usedin another enumeration calculation and proceeds with a new comparisontest. If the sequence number is less than the final tally, it is anindication that the sum of the encoded output sequence is less than thecurrently evaluated sum, but greater than the previously evaluated sum.

The data process then proceeds to decode the sequence number byperforming the same positional notation described previously, startingwith the first member. The process compares the calculated number withthe unique sequence number read. If it is less than the read sequencenumber, the process iterates; it increments the state value of themember being generated, starts a new enumeration calculation and addsthat calculated number to the final total. If the calculated number isgreater than the read sequence number, the process stops and marks themember being generated as a match. The process records this member'sstate value as the value for the first member of the output sequence.The process advances to the next member in the sequence and repeats saidcalculation, incrementing, testing and matching. The result is thedecoding of the original output sequence. Once the generated sequence isdecoded, the data processor applies the inverse of the sequence functionto the generated sequence.

Possible Embodiments

Many variations to the steps described above are possible. For example,the argument values for the continuous functions are generally notrequired to be included with the ordered list of identifiers forrandom-like data as presented in this first example, for the spatialmodel of a random-like data system can be included with the decoder,since the statistics of random-like data is generally uniform for allstates in the state subset, unlike ordered data, which has a spatialcharacteristic that varies across ordered-like states. Where argumentvalues can be useful is when the choice of modeling a smaller area ofthe positive probability subspace is applied. In particular, this iswhen the total number of bits required describing the argument values,the function receiving the argument values, and the ordered list ofidentifiers representing the sequence number is less than the option toencode only the sequence number within a much broader positiveprobability subspace included with the decompressor. In that scenario,it may be beneficial to include the continuous function and its argumentvalues with the list of identifiers. For example, when the two line orcurve boundaries comprising the positive probability subspace creates amuch narrower channel, as the positive probability subspace wouldcontain far fewer sequences, resulting in a smaller sequence number.FIG. 8, in chart 800, shows output sequence 802, enclosed in a narrowchannel by continuous curve 803 and 804, which represent boundaries tothe positive probability subspace 805 and 806 within space 801 of twodimensions; axis x 807 and axis y 808. This model describes a muchsmaller number of probable output sequences within the positiveprobability subspace.

Another possible technique for the selection of constraint elementswould be to define the positive probability subspace by selectingconstraint elements within the space discretely. By traversing allmember state values for each output sequence read and comparing eachalong the time axis (x), the highest and lowest state values at eachinterval on the x axis can be used. These values may act as constraints.

In another embodiment, the selected probability constraints can beembedded in the source code or hardware processor of the decompressoritself and the compressed code simply contains an argument value thatidentifies whether it is a random type data or not. In that embodiment,the argument values of the function need not be included with theencoded message, only a “yes” or “no” to confirm whether it fits thecriteria of a given statistics model.

A number of states within a state subset may also be calculated andstored in an a priori generated set or index. In such an embodiment, theencoder/decoder need not enumerate the number of states within a subsetwhen encoding the sequence number using the positional notation, asdescribed previously. It would simply look up the number of probableoutput sequences in the a priori generated set to encode/decode thesequence number.

In yet a different embodiment, a deviation from the known model can beincluded in the encoded message, whereby a standard constraint set isincluded with the decoder and a deformation value, included with theidentifiers, defines a difference between a known or current argumentvalue and a different or previous argument. If deformation values areincluded with the identifiers, the system will receive them to offsetthe standard argument values or previous argument values of the spatialmodel to determine the probability values of any possible state of thesystemic characteristic. Due to the fact that different models would allhave the same topology, a continuous function may deform the standardmodels to approximate a single sequence or a set of sequences.

Any method or apparatus that selects a set of one or more probabilityconstraints for an enumeration calculation, whereby the constraints arefitted tightly or loosely to an area where a probability distribution ofmembers exists in a structured set or space, and where that area is lessthan the total space, is within the scope of this inventive material.This includes a smooth function that fits the points of another smoothfunction and boundary area, which in turn fits another distribution ofmembers. As subdivision surfaces is used in computer graphics, so toocan the shape of the boundary be subdivided to provide improvements inthe speed of encoding/decoding the output sequence and sequence number.In that embodiment, a lower resolution sequence is refined throughhigher and higher subdivisions until the shape of the sequence matchesthe output sequence at the highest resolution, essentially using ahierarchical multi-resolution structure to the space comprising theoutput sequences. Such methods that may be used for defining the shapeof the data system, output sequence and probability distributions are;continuous functions, splines of various kinds, multi-level subdivisiongeometry, rays, line segments or polygons defined by discrete pointsusing interpolation to smooth the in-between spaces. A continuousfunction is not required to generate a set of constraints. It can,however, be considered an advantage.

The benefit of using a continuous function rather than a more discreteselection technique is that the data required to generate theconstraints for all members corresponding to elements/points in a spaceis lessened when using interpolation. Thus, continuous models can begenerated using less data to define them, rather than if the shape wasdefined discretely. A continuous function can describe the essentialform of the data system represented within a space, which allows thestructure of a data system to be independent from the resolution of theset. A “sufficiently” smooth shape would mean that the probability ofthe systemic characteristic determined by the continuous functionrepresents an infinitely smooth output in comparison to the morediscrete space defined by elements. For example, a spline may create asmooth, interpolated curve with only a few points, when a system cantake the control points and generate the smooth curve. Therefore, thenumber of elements required to describe a form of the state subsetboundary can be less when using smooth, continuous functions. Thecontinuous function need not have control points, but any argument valuethat may allow it to generate its output.

Regarding the step of calculating the number of probable outputsequences within the positive probability subspace, the use ofinclusion/exclusion does not limit the scope of the inventive subjectmatter. Other methods to determine the number of states within the statesubset may be used. Algorithms or functions for calculating the totalnumber of sequences contained by said constraints may be found in theknown art and be the following; Dynamic programming, GeneratingFunctions, Summation Functions, Enumeration functions and StandardDeviation functions.

A dynamic programming function may be as follows:

With nonnegative integers Q_(i), j=1 . . . n, let F(y,q) be the numberof solutions of x1+ . . . +x_(q)=y with x_(j)ε{0, . . . , Q_(j)} foreach j. Then F(y,1)=1 for y=0, 1, . . . Q₁, 0 for y>Q₁, andF(y,q+1)=Σ_(min(y,Q) _(q) _(+1)x=0)F(y−x,q).

A generating function may be the following:

(1+x+ . . . +x^(constraint) ^(_) ^(a))(1+x+ . . . +x^(constraint) ^(_)^(b))(1+x+ . . . +x^(constraint) ^(_) ^(c))(1+x+ . . . +x^(constraint)^(_) ^(d)) . . . , where the generating function is able to calculatethe number of possible sequences given the constraints, which is theexponent of x.

A summation function may be the following:Σa=highconstraint_aΣb=max(a+1,lowconstraint_b)a+highconstraint_bΣc=max(b+1,lowconstraint_c)highconstraint_cΣd=max(c+1,lowconstraintd)highconstraint_d. . . X.

The summation equation sums all the possible sequences within the highand low constraints that equal a given total. This is essentially anenumeration function that counts all possible permutations of asequence, sums the members and tests whether the total equals one of thetotals within the positive probability subspace.

A standard deviation function would take the total number of allprobable states within a state subset and divide that total along theconfines of the normal distribution model, which is shaped like a bellcurve. Thus, the first total within the normal distribution may be 1,where the members of the sequence sum up to the lowest possible total.The number of possible sequences summing to subsequent higher totalsincrease, matching the slope of the normal distribution until it reachesits zenith, at which point the number of possible sequences diminishesas each total at the other side of the bell curve is evaluated until thenumber of sequences at the highest possible total is 1. The function maydetermine the number of possible sequences summing to any particulartotal based on the normal distribution.

Another possible technique for determining the number of probablesequences obeying constraints may be by treating the space as a lattice.The constraint elements may be constraints for a monotonic lattice path.The total number of possible lattice paths is calculated using thebinomial coefficient

$\quad\begin{pmatrix}{x + y} \\x\end{pmatrix}$

where x and y are the maximum size of the dimensions of the lattice. Thefirst operation subtracts all of the possible monotonic paths that passthrough one of each constraint. The same binomial coefficient formula isused to calculate from the constraint point to the next point in thepath, which in the first series of calculations is the end point or theupper right corner of the domain. The product of these binomialcoefficients equals the total number of possible monotonic paths fromthe origin, through the constraint point(s), to the end point of thelattice. Each of these for every constraint are summed, and thensubtracted from the total number of possible paths without constraints.Using the principle of Inclusion/Exclusion, the first set ofcalculations over counts the number of possible lattice paths that weresubtracted. It must add back the total number of possible monotoniclattice paths that pass through two pairs of constraints in the nextseries of calculations. These multiple series of calculations repeats,alternating from subtraction to addition of constraint sets until allcombinations for the number possible monotonic paths passing through theconstraints are counted. The final result is the number of monotonicpaths that do not pass through the constraints.

Any method or apparatus that calculates or estimates a total number ofprobable states or sequences within a state subset of a data systemfalls within the scope of this inventive material.

As stated previously, the number of probable sequences can be reducedfurther using a spatial statistical model where a sequence functionprocesses an ordered sequence and outputs an output sequence where thestate values of the members signify a measure of a variance from adetermined value, such as the mean, median or mode of a sequence. Suchis the case where a total sum for 16 members is 2032 and, dividing 2032by 16 yields 127, which makes “127” the mean value. A measure of theordered sequence's deviation from the value 127 can be expressed by theoutput sequence.

FIG. 9, in diagram 900, shows the members representing the deviationfrom the mean value 127, which can be accomplished by putting themembers 903 in lexicographic order. The members are bounded by boundary904, of positive probability space 905. A second monotonic sortingoutput sequence representing a sequence out of a total of 16! possiblesequences may be employed to sort the lexicographic ordered sequenceback into an unsorted sequence, as described previously. A series ofmonotonic sorting sequences are shown in the two dimensional space 907,using x axis 908 where the intervals are normalized from 0-100% and,with y axis 909 representing the state value from 0-130, for eachsorting value is summed by the previous sorting value in the outputsequence using a summation sequence function or an monotonicallyincreasing function. The probability field is defined by boundary 911.The members of the output sequence 910 are enclosed within positiveprobability subspace 914, which is bounded by constraint boundaries 912and 913. The zero probability subspaces 915 and 916 are also displayed.The number of states of each subsequent member decrements by 1 from anumber of 16 possible states (i.e., 16, 15, 14, 13 . . . ), if each ofthe 16 members in the lexicographic ordered sequence has different statevalues. If two members in the lexicographic ordered sequence have thesame state value, then the highest possible state a member could bewould be 15 rather than 16.

The variance from the mean can be exemplified by the degree of the slopeof the boundary curve in space 900. The smoother and more pronounced theslope of the boundary containing the positive probability subspace ofthe lexicographic ordered sequences, the greater the variance betweenhigh and low values of the output sequence. For the case ofpseudo-random or random-like data, the likelihood of each member havinga value of 127, as exemplified by the upper most boundary 902, or wherehalf the members will have a state value of 0 and the other half with astate value near 255, is not probable or within the positive possibilitysubspace, 905. Using this technique can further reduce the number ofprobable sequences having a sum between the ranges of 600-3420.

The sequences from a data system may also be modeled using variouslengths, as can be shown in FIG. 10, which is a space 1000, with anormalized state dimension, y axis 1001, and a time dimension on x axis1008. The boundary 1007 comprises the space that represents the set ofall possible output sequences (or all possible states of a data systemof a given length/size) where the sums of the members are normalizedwith a value from 0 to 1000. Each set of output sequences comes from thesame data system in a state of random-like data, which is over 1,000,000bytes in length/size. The FIG. 10 shows how the output of the sequencefunction is displayed in a chart, the amount of bytes it processesdoubles for each set of output sequences. The first set of outputsequences, 1002, is 16 bytes in length. The second set of outputsequences, 1003, is 32 bytes in length. The fourth set of outputsequences, 1004, is 64 bytes in length. The fifth set of outputsequences, 1005, is 128 bytes in length. The sixth set of outputsequences, 1006, is 256 bytes in length.

There are a wide variety of embodiments that are possible with regardsto how the space is structured and how a sequence function performs; thetwo being a way of determining the state subset of a data system using aspatial statistical model. In addition to the way a space is structured,there are many possible sequence functions as described previously.Thus, the possibilities of using a model should be viewed to itsbroadest extent, for there are many ways of modeling a data systemwithin a space. The space can conceptually have only 1 dimension, if thetime dimension has no intervals. There can also be three or more statedimensions to represent a multidimensional space. The aspect of thisinventive material covers any kind of set with added structure, such asEuclidean, Non Euclidean, phase and topological spaces that have atleast one dimension. As stated above, there may be many types ofstructures and relationships that may be added to a set. Anyrelationship, structure and function added to a set may apply.

As stated previously, using a space is not a limitation of the inventivesubject matter for determining the state subset. The state subset of adata system can also be, at least partially, determined based on anyprobability of a system's state and other characteristics. For example,when the probabilities of the frequencies of the states of one byte areknown, as for a sequence comprising 1024 bytes, an enumerationcalculation can determine a number of sequences that obey thatprobability distribution. For example, if the probability distributionassigns relative equal probabilities to the frequency of the 256possible states of one byte, then using an enumeration calculation, suchas the inclusion-exclusion algorithm described earlier, can determinethe number of sequences comprising bytes where all its possible statesappear with relative equal frequency. Using other features of thepresent method, set of machine instructions and apparatus, such as aprobability field and constraint elements, offer greater advantages,such as a determination of a state subset's boundary, form and structurewithin a one or multi-dimensional space.

The next example will apply the techniques described above for thecompression of a larger random-like data file.

Second Example Embodiment

The second example of the preferred embodiment will use the same spatialstatistical model and argument values for random-like data described inthe first example to compress two random-like files of 4,096,000,000bytes into a single file using an iterative compression technique. Italso explains how the files may be encoded and decoded using a tree datastructure, which allows the method, article comprising machineinstructions, and apparatus to decode only selected files withoutdecoding all files.

All the steps described in the first example for determining a model fora data system's state subset also applies to the second exampleembodiment, which includes: establishing a symbol set, receiving datasystems of similar systemic characteristics, accessing a sequencefunction that operates on the ordered sequence from the data system andgenerates an output sequence, constructing the space, corresponding themembers of the output sequences to the elements of the space,determining probability values for the possible states represented bythe elements of the space by various means, such as selecting elementsas constraints. The same spatial statistical model for random-like dataused in the first example is used to determine the state subset thatwill allow for the compression of two 4 gigabyte random-like data files.The symbol set criteria and the sequence function is identical to thefirst example; the data is read as a series of sequences 16 bytes long,with all 256 states of the 8-bits being considered. The description ofthe second example begins at the stage of constructing the ordered listof identifiers, which represent all the probable 16-byte sequenceswithin the positive probability subspace of the spatial model.

As shown in FIG. 11, the set of possible identifiers 1100, may includethe following in this present second example embodiment:

-   -   1. Symbol Set criteria 1101 (if required).    -   2. A state subset type 1102 (a random-like state subset type)    -   3. A deformation value 1103 from a previous probability        distribution.    -   4. Size of the data system 1104.    -   5. The tree data structure 1105, including the number of        recursion passes performed on the data (if required).

For each 4 gigabyte file in this example, a series of 256,000,000sequence numbers must be calculated, (4,096,000,000 bytes/16 bytesequences=256,000,000 sequences). When each sequence number isrepresented by its own list of identifiers, the overall code becomesinefficient. This is due to the fact that sequence numbers that cannotbe represented by a whole number of bits (i.e. 5.345 bits) must berounded up to the nearest whole bit to be received by the dataprocessor. To ensure the code is as optimal as possible, the method,article comprising machine instructions, and apparatus can create onenumber that represents all sequence numbers and argument values. Thiscan be achieved using a mixed-radix positional notation system whereeach sequence number from the set of sequence numbers becomes the radixof a larger number. This is allowed due to the fact that the maximumranges of the argument values and sequence numbers are known oncecertain aspects of the message are known and decoded first. For example,when the process knows the following; the size of the data system, thesymbol set criteria and whether the data system matches a random-likedata statistical model, the system is then able to determine the maximumsize of the ordered list of identifiers, or the highest possiblesequence number that can represent an output sequence bound by thespatial model.

To generate this mixed-radix number, the system multiplies the maximumranges for all of the radixes preceding a selected radix, and thenmultiplies that product by the value of the selected radix. The resultis added to a Final number. The process moves down to the next radix,repeats the same process described above, and then adds that number tothe Final number. For example, FinalNumber=(n×(r1×r2×r3))+(n2×(r1×r2))+(n3×r1)+(n4), where n is the sequencenumber and r is the radix signified by the equations in eachparenthesis.

The result is a coded number, which can be uniquely decoded using themixed-radix positional notation. This single number, as well as theother aforementioned values that were not encoded into the mixed radixsystem, are added to the ordered list of identifiers.

Using the mixed-radix number system, 10,000,000 sequences with 16members, or 1,280,000,000 bits can then be expressed using 1,279,999,996bits or 159,999,999.5 bytes (127.9999996*10,000,000=1,279,999,996 bits)using the spatial statistical model for random-like data. The term“compressible data block” is used herein to mean a size of data thatwill result in data compression using the data processing and encodingtechniques of this inventive material.

Data processing systems usually process information in bytes. If thecompressed message cannot be represented using a whole number of bytes,as is the case for 159,999,999.5 bytes, two methods may be used out ofthe possibilities contemplated. First, the system can calculate a numberof compressible data blocks that, when each block is appended as onedata system, will result in a whole number of bytes. The method can thenprocess that number of compressible data blocks from the data streamreceived by the encoder. This would result in an encoded region and anon-encoded region. The term “encoded region” or “encoded data blocks”is used herein to mean data, or compressible data blocks, encoded orcompressed within the context of the techniques described in thisinventive subject matter. The term “non-encoded region” is used hereinto mean the bits of the original data system (or data system from aprevious iteration) that did not go through the encoding/compressionprocess in the current transformation cycle. For iterative encoding, thebits in the non-encoded region are as bits in a queue that is suppliedto the encoded region when needed. A second possible method may add“buffer bits” of a quantity from 1 to 7 bits at the end of the encodedmessage. The term “buffer bits” is used herein to mean bits that areadded at the end of the encoded message to “fill-in” for the 2^(nd)through 8^(th) bits that are vacant in the last possible byte of thenon-encoded region.

FIG. 12 with diagram 1200 shows how this process may be applied for oneof the 4 gigabyte files. The 4 gigabyte file 1201 is represented as agrey block, which is then split up into 25 compressible data blocks,such as block 1203. The process of determining these blocks isexemplified by arrow 1202. Because a limited number of compressible datablocks can fit the size of the data system or file, some of the datacannot be encoded as a compressible data block; it is thereforedetermined to be data within the non-encoded region 1204. Thecompressible data blocks are compressed, denoted by compression arrow1207, in a 1^(st) encoding pass exemplified by encoding arrow 1205; thecompressible data blocks become encoded data blocks 1206. Thenon-encoded region 1204 is appended to the last block of the set ofencoded data blocks 1206. If the encoded data blocks and non-encodedregion do not add up to a full number of bytes, buffer bits 1208 areappended to the end of the non-encoded region. FIG. 12 also displays howthe data is processed using more than one compression cycle, which isdescribed in the following paragraphs.

For this present example, as seen in FIG. 12, each encoded data block isappended to a preceding encoded data block. For example, the last 4 bitsof the first encoded data block is filled in by the first four bits ofthe second encoded data block, thus a bit-wise shifting operation isrequired to process the data, which results in 4 buffer bits at the endof the last data block. Because a typical data processing system onlyreads/writes bytes, there cannot be a byte less than 8 bits. Therefore,buffer bits can be appended at the end of the encoded message if thenumber of bits does not add up to a whole number of bytes. If theencoded data blocks are appended in the way described above, only 4buffer bits will be required to be appended at the last bit, giving anet compression equaling N number of compressed bits minus the number ofbuffer bits.

For example, 127.9999996 bits multiplied by 10,000,000 equals1,279,999,996 bits, which is 4 bits less than 1,280,000,000 bits or160,000,000 bytes. Therefore, a compressible data block for eachrandom-like data file of 4,096,000,000 bytes, amounts to 160,000,000bytes that can compress by 4 bits. There are 25.6 compressible datablocks in each 4 gigabyte file. In this second example, only fullcompressible data blocks are accounted for. This gives 25 fullcompressible data blocks, as depicted in FIG. 12 as the set ofcompressible data blocks 1203; each 4 gigabyte file being compressed by100 bits or 12.5 bytes, the compressed file being 4,095,999,987.5 bytes.The result gives a total of 25 bytes being compressed out of the twofiles using the techniques of the present method, article comprisingmachine instructions, and apparatus as presented in the first and secondexample embodiments. The fractional data block of 0.6 for each file (12Mbytes) is determined by the processor to be data within the non-encodedregion, as depicted in FIG. 12 as dark grey blocks 1204, 1211 and 1215,which are appended to the last bit of the last encoded data block.Because 12.5 bytes have been compressed out of each file, 4 buffer bitsare appended at the end of the non-encoded region to eliminatefractional bytes, netting a total of 12 bytes being compressed out ofeach file. This is shown in FIG. 12 as buffer bits 1208 and 1218.

One benefit to the present method or apparatus of this inventivematerial is the ability to iteratively compress the data if the outputis also random-like data. This is possible when the same spatial modelfor random-like data can be applied for each transformation cycle. Thisis a unique advantage to using random-like data, which has a generaluniform model for all random-like states. The following is a descriptionof how to recursively apply the techniques described above.

While it may be probable that the list of identifiers generated by themethod or apparatus may also embody the systemic characteristics ofpseudo-random or random-like data, it is not impossible for the encodedmessage to also have systemic characteristics of ordered-like data. Thisis because the spatial model is intended to contain all possiblerandom-like sequences, therefore all possible states within the statesubset, and all possible sequence numbers representing those states, areconsidered equally likely. To apply compression iteratively, thesystemic characteristics of the incoming data system must match thestatistical model for the recursive technique, which in this case israndom-like data. To ensure that this occurs, a random filter functioncan be applied to the bits representing the ordered list of identifiersto ensure it is random-like. This process can be accomplished byapplying a XOR bit-wise operation on the bytes of the encoded messagewith a known data system file of the same length and in the random-likestate. When the bit pattern of the random filter file is known to thedecoder, the original file can be decoded by applying the inverse XORbit-wise function using the known random filter file. This random filterfile need not be included in the encoded message nor be of anysignificant size, as it may already be included with the decoder andapplied for each transformation. Once the encoded message has beentransformed to a random-like state, the process can input thetransformed encoded message as a new data system to be received by thedata processor, which initiates the start of new iterativetransformation cycle. The process writes a number of transformationiterations performed within an index.

Iterative Encoding and Tree Data Structure

A tree data structure for the iterative encoding/decoding allows thedata processor to decode selected appended files without requiring thedecoding of all compressed files merged in the process, such as thosebelonging to the second random file. The system counts the number ofrecursion passes before the two files merge in the tree. A node isrecorded whenever a recursion path or a data system being encodedappends another data system.

For example, each of the two random-like data files will be encodedusing several recursion passes until the data can no longer becompressed efficiently due to its size, which may be a size matching acompressible data block, as depicted by the encoded data block 1217 inFIG. 12. When this occurs for the paths of each random-like data file inthe tree data structure, the system can append one data file to theother, merging them as one data system, and continue the iterativeencoding further. The number of iterative passes, as well as the nodesindicating when two or more data systems are appended, is recorded inthe tree data structure.

When the 4 gigabyte random-like data file undergoes a second recursiveencoding pass, the number of bytes for each file is reduced to4,095,999,975 bytes. The number of whole compressible data blocks isstill 25 for each file at the second iteration, reducing the data againby 100 bits or 12.5 bytes and, appending another 12 Mbytes of the fileto the non-encoded region as was done in the first encoding iteration.Because 12.5 bytes were compressed out of the first and second recursioncycle, the total number of bytes compressed out equals 25 bytes for eachfile, allowing for the 4 buffer bits added to the file in the first passto be eliminated. This operation is illustrated in FIG. 12, which showsthe 1^(st) encoding pass represented by the arrow 1205 and the resulting25 encoded data blocks 1206 that are each shown to be smaller than thecompressible data blocks 1203. A new set of compressible data blocks1210 are determined from the encoded region 1206 and non-encoded region1208 from the 1st encoding pass 1205, all of which is considered a newdata system to be received and encoded by the data processor in thesecond encoding pass. Multiple encoding passes may follow the second, asillustrated in FIG. 12; the multiple encoding passes arrows 1213 and1216, along with the compression arrows 1207 and 1219 illustrating theprocess that compresses the data to 10 compressible data blocks 1214,along with a non-encoded region 1207, and then to only one encoded block1217 with buffer bits 1218 appended. In this present example embodiment,iterative encoding is unable to compress data beyond the size of acompressible data block.

FIG. 13, 1300 shows how iterative transformation/encoding cycles may beconstructed within a tree data structure. Three data systems 1301 areappended in an appended data system represented by node 1303. Otherappended data systems up the tree are also represented by nodes 1307 and1311. Paths 1302, 1306 and 1310 represent where data systems areappended and thus are connected to a node. A node may also mark thebeginning of a series of iterative transformation/encoding cycles 1314,1308 and 1312, where the appended data system is received by theencoding method, set of machine instructions and apparatus. A series ofprocesses (and means for performing the series of processes) areexecuted in a transformation cycle, as shown in block 1314. An index1304 is added to the tree to represent the number of occurringtransformation cycles. Each time a transformation cycle is performed,the number comprised by the index is incremented. There may be a seriesof transformation cycles before the data system reaches a size where itmay be appended with other data systems or files 1315 and 1316 into anew appended data system, which is seen in FIG. 13 as node 1303, 1307and 1311 up the tree at branches 1302, 1306 and 1310. Using a tree datastructure for iterative encoding/decoding allows the present inventivesubject matter to decode parts of the data, rather than an entire set ofcompressed data systems of files

As presented in the present example embodiment, a number oftransformation iterations can increase until the compressed code is atleast the size of one compressible data block. This does not includeinformation describing the tree data structure for recursive encoding orbuffer bits. The size in bytes of the tree data structure depends on thenumber of branches or merging of data systems during the iterativeencoding and the number of iterative passes in-between those mergingevents or branches.

If the two random files were in a folder of a filing system, then thesystem would include another tree data structure, specifying the twodifferent files, file names, file size and any other data necessary todistinguish the two files in the context of a file system of a computeroperating system.

The decoding process is the same as described in the first example,which is essentially reversing the process of encoding the members usingthe numeral system. The difference in this example is that the processrequires the decoding of the mixed-radix number and the use of a treedata structure.

The first step is to read the correct number of bits to calculate thenumber generated from the mixed-radix numeral system previouslydescribed. This is accomplished by receiving the spatial statisticalmodel and calculating the maximum number of sequences in the positiveprobability subspace within the constraints based on the size of thedata system included with the identifiers, as described in the firstexample. When the maximum number for the sequence numbers of each radixin the mixed number system is known, the coded number for each radix canbe calculated as follows: if the maximum for each radix was 100, n canbe found using the following equation: n=FinalNumber/((n×(r1×r2×r3))+(n2×(r1×r2))+(n3×r1)+(n4)).

Because the second encoded data block is appended at the 5th bit of thelast byte of the first encoded data block, the method or apparatus willneed to shift the bits using bit-wise operations in order to read thedata. Among the possibilities contemplated, this can be done by readingthe necessary bytes for the first encoded data block, removing the lastfour bits using bit-wise operations and calculating the second encodeddata block from the remaining bits. The last four bits from the firstencoded data block must be removed and shifted so that they are thefirst four bits of the second encoded data block. The last four bits ofthe second encoded data block are shifted at the end. Of course, thedescription of using bit-wise operations is not a limitation of themethod or apparatus.

If a tree data structure was used in the transformation/encodingprocess, the generation/decoding process can access the same tree datastructure included with the ordered list of identifiers to only decodeselected data files/systems. This is exemplified in FIG. 14. 1400.

In order to generate/decode a selected data system transformed/encodediteratively within a tree data structure, the data processor must firstcalculate a shortest path between the compressed data system or rootnode 1401, and the selected data system or child node 1415 selected bypointer 1417. A method in the known art for calculating the shortestpath for a set of nodes can be the A* search algorithm described inHart, P. E.; Nilsson, N. J.; Raphael, B. (1968). “A Formal Basis for theHeuristic Determination of Minimum Cost Paths”. IEEE Transactions onSystems Science and Cybernetics SSC4 4 (2): 100-107. When the shortestpath is determined, as seen in FIG. 14 as shortest paths 1403, 1408 and1413, the data processor accesses only the nodes comprised by thisshortest path, such as nodes 1419, 1420, and 1421, which represent theappended data system. The compressed data system 1401 may then undergothe series of generation cycles 1418 based on the index of the number ofoccurring generation cycles 1402 until node 1419 is accessed. The dataprocessor may then separate the appended data system into separated datasystem or file 1406 and 1405. All separated data systems are attached toa node by a path, such as paths 1403, 1404, 1408, 1409, 1413 and 1414.The system may delete separated data system 1405, 1410 and 1416 notattached to the shortest path. Deleting separated data systems 1405,1410 and 1416 can save data processing system's memory and storageresources when iterative decoding is implemented using a tree datastructure. The data processor repeats the procedure for the next datasystem 1406 and 1411 down the tree. They too undergo the same processfor the series of generation cycles 1407 and 1412 until data system 1415is decoded. The tree data structure allows a data system selected withinthe tree to be decoded without decoding all data systems 1405, 1410,1416 and the many others.

Among the possibilities contemplated, the above embodiment may beexpressed using various approaches. For example, in another embodiment,rather than the data elements signifying raw bytes in a data system, atechnique can signify the data elements as frequency counts and timemarkers along the length of an ordered sequence from the data systembased on 8-bit byte patterns. The term “Frequency count” is used hereinto mean a value that describes the number of times a data unit, a bytein this case, occurs along a sequence of ordered data units. The term“Time marker” is used herein to mean a value that describes the numberof data units, or bytes, that occur before or after a distinctive dataunit, such as a chosen 8-bit pattern. For example, if the distinctivedata unit was the byte “8”, a time marker value of “80” would describe“8” appearing on the 80th element in a data system.

The method or apparatus can read the incoming data stream in chunks of4096 bytes and count the number of times each possible byte appears.These frequency counts can be plotted in a space where each possiblesymbol corresponds to a unit along the x axis, for example, from 0-255for 8-bit patterns. For the y axis, each unit signifies a frequencycount. For example, if byte “0” occurred 13 times, the first memberwould be assigned the value of (0,13) x,y respectively. A monotonic pathwould result in the sum of all members adding to 4096. The members mayalso be sorted in lexicographic order.

The time markers can be plotted in the space as follows; the x componentsignifies the time dimension or the number of symbols along the datasequence from 0-4096; the y component signifies the time marker value orwhere a particular symbol occurs along the sequence of bytes. The “timemarker” members can also use a sequence function to sort it inlexicographic order. The argument values of a sorting sequence functioncan be passed to the inverse of the sorting function to sort thelexicographic ordered sequence back to its original ordered sequence,whereby the states of the argument values are plotted in a differentspace.

Spatial probability models would be determined for the sequence ofargument values as exemplified in the first example that was previouslydescribed. Probabilities would be determined using constraints and/orprobability distribution functions. Constraints can be used in thecalculation that enumerates all the sequences within the positiveprobability subspace. Variable-length encoding can also be employed witha probability mapping or probability distribution function.

To decode the message, the data processing would read the sequencenumbers, which are the frequency counts, the time markers and theargument values for the sorting function using the techniques previouslydiscussed in the first example. Knowing the frequency count for aspecific byte symbol and the respective time markers allows themethod/apparatus to write the correct number of bytes at the correctlocation along the sequence of 4096 bytes. This can be done for eachbyte symbol.

This example is to show an alternate way of utilizing the techniquespresented in the inventive subject matter, showing how they may beapplied using a variety of techniques, such as using different symbolsets and criteria, sequence functions, and various kinds of spaces ofone or more dimensions for data processing. The methods presented in thefirst and second example embodiments cannot be limited to a type ofdata, symbol set, sequence function, enumeration calculation,variable-length encoding method, and space.

Third Example Embodiment

The third example describes a way to use a probability distributionfunction and a probability mapping function to determine probabilitiesfor states within the probability field, as well as incorporatingvariable-length codes into the ordered list of identifiers.

The steps for constructing the spatial statistical model described inthe first and second examples also applies in this third example; a datasystem is received by the data processor, the sequence function receivesan ordered sequence from the data system and the output sequence isgenerated, the space is constructed where the members of the outputsequence correspond to the elements of the space and the distribution ofthe members in the space are fitted by probability constraints. Todetermine the probability values more accurately for the spatialstatistical model, the process determines a continuous gradient functionthat maps a smooth, continuous distribution of probabilities to theprobability field comprising state representations or elements,particularly within the positive probability subspace.

A variety of techniques in the known art for generating a gradient canbe applied to the elements of the space. A process can iterativelycompare the output of a continuous gradient function generated withinthe positive probability subspace, based on the probability values ofthe possible states. For example, a continuous gradient function isexemplified in FIG. 15, where space 1500 has an x axis, 1503, and a yaxis, 1507. The boundary 1508 contains the probability field. Theconstraint boundary 1504 and 1509 contains the positive probabilitysubspace 1505 where members of the output sequences correspond toelements. The gradient 1502, generated by a probability distributionfunction, is contained in positive probability subspace 1505, where line1506 represents the mean of all output sequences, which is used todetermine the beginning of the gradient as it spreads out towardconstraint boundaries 1504 and 1509. A continuous probabilitydistribution function generates a smooth gradient based on the number ofcorrespondences of the output sequences with the elements of the space.

The method iterates through the high and low values of its argumentvalues that controls the falloff of the gradient and checks the outputof the function at each iteration for a “best fit” output that mosteffectively approximates the probability distribution of the members ofthe output sequences. These argument values for the continuousprobability distribution function can be included in the compressed codeor, included with a decoder.

Among the possibilities contemplated, the following is an examplealgorithm for determining a probability assignment for the elements ofthe spatial model using a probability distribution function and ahierarchical multi-resolution structured set. The method constructs andstores a lower resolution version of the space in computer memory, basedon the number of units along the axis of the time and state dimensions.For example, 1 element of a lower resolution space may refer to 5 or 10elements of a higher resolution space. The method then iterates througheach member and checks the elements of the higher resolution space itcorresponds to. For example, if a member corresponded to element (52,35), x and y respectively, on a high resolution space, then the memberwould correspond to element 5 of the time dimension and 3 on the statedimension, (5,3) of the lower resolution space. A numerical strengthvalue would be added to that square within the grid in computer memoryeach time a member of an output sequence corresponds to a lowerresolution point, similarly to the process of adding a value of 1 to thecorrect element corresponding to a member of the output sequence, asdescribed in the first embodiment. The more members that correspond witha point in the lower resolution space, the greater the probability valueof the state represented by the elements in the related higherresolution space. When all the members of all the output sequences havebeen corresponded to the right elements, the result is a probabilitydistribution as defined by the values of each square of the lower gridor space. When a gradient is generated by a function using the lowerresolution space, a courser gradient results. Similar scalar values inthe courser gradient may be normalized or averaged to create a smoothgradient in the higher resolution space. This example within thispresent embodiment is only one of many possible methods that may beutilized to generate a probability distribution gradient based on thecorrespondences of a member to an element, which, as may be appreciatedby one skilled in the art, does not limit the scope of this inventivesubject matter. It should be noted that the elements of the space canapply to a variety of applications, such as; pixels of an image, pointsof a metric space, voxels in a volumetric space, vectors of a vectorspace and so on.

FIG. 16 shows an exemplary diagram 1600 of an encoding process utilizinga variable-length code based on the probability values determined forthe possible states represented by elements in space. A continuousprobability distribution function creates a probability gradient for thepositive probability subspace 1608, which is comprised by boundaries1603 and 1604. The zero probability subspace 1602 is within boundaries1601 and 1603. The probability field, which contains both the positiveand zero probability subspaces, is comprised by the probability fieldboundary 1601. In this example, all elements of the positive probabilitysubspace, which represent a possible state of a member, are assigned aprobability value from the gradient output of the continuous probabilitydistribution function. Members 1605, 1606 and 1607 correspond toelements of the set with added structure; therefore, each probable statefor the member corresponds to the probability value determined for thecorresponding element. This probability value can then be used togenerate a variable-length code represented by the ordered list ofidentifiers using techniques in the known art, such as Huffman andArithmetic encoding, as seen in the diagram as code 1609 and 1614. FIG.16 displays the correspondence 1613 between a probable state 1611 formember 1607 and a variable-length code 1614 within a set ofvariable-length codes 1612. Probable state 1611 and other probablestates for member 1607 are considered to be comprised by its own statesubset 1610. It is to note that the positive probability subspace 1608represents the larger state subset of the data system, while the smallerstate subset 1610 for member 1607 is comprised within the positiveprobability subspace. This example shows how the method, articlecomprising machine instructions, and apparatus may model a smaller statesubset comprised by a larger state subset of a data system's statesuperset. In other words, the data processor may also determine thestate subset for a unit of the data system, such as a member of anoutput sequence.

Out of the many possibilities contemplated, the method can pass anargument value to the continuous probability distribution function toobtain the probability value at a discrete point without constructingthe set with added structure. For example, the method, articlecomprising machine instructions, and apparatus can pass the argumentvalues of the time and state dimension (x,y) to the continuousprobability distribution function, which then outputs the probability atthat coordinate.

While a continuous probability distribution function may be used toassign probability values to each of the possible states represented bythe elements of the space, among the possibilities contemplated, adiscrete probability distribution function and probability mappingfunction may also be used for this purpose. The data processing device,by the data processor, may record the probability values determined foreach possible state using a discrete probability distribution or amapping function that uses a reference to assign the probability to theelement discretely. A discrete probability distribution and probabilitymap can be preferable if a probability distribution of the data systemis determined to have a rather complex or ordered characteristic patternthat is difficult to approximate using a continuous gradient. To employa discrete probability distribution function would be similar to thecontinuous version, except that instead of generating a smooth gradientby interpolation, the function would assign a probability to theelements discretely. To record a probability map, the method can analyzethe number of members that correspond to each element of the space anddetermine a probability value by dividing the number of correspondingmembers at an element in the space by the total number ofcorrespondences. For example, if there were 1000 output sequences, andan element corresponded to 500 members, a probability value of 0.5 wouldbe assigned to that element and recorded for the probability map. Thiswould be calculated for each element of the space when a probability mapis used. When implementing the spatial statistical model for dataprocessing, the method can access the probability map and assign theprobability values to the elements, based on the argument values.

Among the possibilities contemplated, the method can also allow thedesignation of multiple probability distribution functions or maps basedon relationships between members. This results in a variable-length codebased on multiple probability functions/maps, where the probabilityvalues for each element are based on a relationship between a currentlyevaluated member and a member that precedes the currently evaluatedmember in the output sequence. To accomplish this, the spatialstatistical model is used to determine and record the probabilitiesbased on the patterns of two are more members connected by arelationship, resulting in the generation of a probability distributionfor all possible relationships between the members. FIG. 17 shows agraph of how this can be accomplished. The space 1700 and the boundaryof the probability field 1701, contains positive probability space 1706constrained by boundaries 1703 and 1705, with the zero probabilitysubspace 1702 being comprised by probability field boundary 1701 andboundary 1703. A first member 1707 of an output sequence has a statevalue of “156”, followed by second member 1708 with a state value of“81”. In this case, the probability map 1712 attributed to the secondmember 1708 is dependent on the state of the preceding member 1707.These dependencies are shown in FIGS. 17 as 1710 and 1711. Probabilitymap 1712 is dependent on the state value of the first member 1707. Dueto the fact that there are 256 possible states (0-255), there are 256possible probability maps that may be assigned to elements correspondingto the second member 1708. In FIG. 17, probability map 1712 is labeledas “Map 1.156”, which is to signify that it is the 156^(th) probabilitymap in a map set assigned to the second member 1708. The secondprobability map 1713 is dependent on the state of the first and secondmember. This second mapping has a 65,536 possible probability maps thatcan be assigned to the elements corresponding to the third member 1709.A variable-length code is generated based on the probability value, asseen for the variable-length code 1708 that is assigned to the thirdmember 1707. A set of variable-length codes 1714 is based on theprobability values assigned to the elements by the probability mapfunction 1713. One thing to note, probability map 1712 and 1713 each area representation of a state subset comprising the possible states ofmembers 1708 and 1709. While the states of the members, as a wholesequence create a state, the individual members themselves also havetheir own state subsets, as seen in FIG. 17.

Out of the many possibilities contemplated, the method can also includemore than one set with added structure, dimension or member to multiplythe number of possible states that can be expressed, which may also beassigned probability distribution function or maps based on thoserelationships. The point of this particular example is to show what ispossible to implement when the present method, article comprisingmachine instructions, and apparatus utilizes more than one probabilitydistribution function or map for the space based on a variety ofpatterns, relationships and conditions used with variable-length coding.

Essentially, the variable-length code can depend on more than oneprobability distribution function, map or space. If there are more thanone probability maps or functions associated with a state, thevariable-length code will incorporate the probability values of themultiple probability functions/maps for the encoding of a member. Theaspect of the present inventive material that should be clearlyunderstood is the capability of using a function to generate and assignthe probabilities values to the elements. Any process that determines aprobability for the possible states of a systemic characteristic lieswithin the scope of the present inventive subject matter.

For the process of encoding/decoding, the first step is to generate theprobability gradient that determines probability values for the possiblestates and implement the spatial statistical model. The second step isto utilize a variable-length encoding method which bases the code onsaid probability values. For decoding, the process reads the list ofidentifiers or variable-length code and references that code to thestate or element of the space in the model using the same probabilitydistribution function that generates the gradient in the space bypassing the argument values of the model to the function, whereby thefunction outputs the probability of the state at that location in theprobability field. The member of the output sequence is constructed bymatching the code/identifier to the probability value, which is assignedto a particular element representing a state. When the match iscomplete, the correct state of the member is deciphered. An importantaspect of the present inventive material is that these probabilitiesneed not be described discreetly or adaptively. According to the presentembodiment, once the probabilities are known from the output of thefunction or map, all that is needed is a method that refers thevariable-length code to its respective value on the y axis or statedimension. The advantage to using this technique is that no enumerationcalculation is required to encode/decode the output sequence. Anotheradvantage is that the probable sequences can have variableprobabilities, thus making possible a more efficient description of theprobable states using the variable-length code.

Use Cases

Data Storage: Data storage involves data backup, recovery and archival.In most cases, data is stored on a secure tape medium for long term backup. The challenge involved with data storage is that the amount of datato archive securely, many times at a remote location, can be large.Compressing data usually results in data that has high or maximalinformation entropy, thus having random-like characteristics. Iterativeencoding/decoding of compressed, high entropy data, as exemplified inthe first, second and third example embodiments, can be used to increasethe compression ratios, which can be a huge advantage for data archival.There would not be as much compressed data to a storage media such astape using the iterative encoding method presented in this material. Ifa tree-data structure is used in the iterative encoding, it will allowcertain files of the data to be recovered without decoding all the data.For example, a database would be compressed iteratively into a smallerfile. A single tape would then be able to store a large amount ofiteratively compressed data. If a particular file needed to berecovered, then the tape would be read and, using the iterative treedata structure technique presented in the second example embodiment, canrecover only the desired file without needing to decode the entire setof files compressed. This allows for efficient data compression andencryption for the purpose of data storage.

Database machines/services: Database machines and services requiredynamic access of files to and from a database device. They also requirea large number of fast storage media to store and retrieve the data forclients connected to a network. The need for using a tree-data structureusing the iterative encoding presented in this inventive material allowsdata over networks, such as websites, games, videos and applications, tobe accessed without decoding all the data, resulting in a fast andefficient method for accessing files from a database machine. This isparticularly important for multimedia services over the Internet, suchas streaming video, audio and interactive media, video games orapplications. For example, a client connects with the database andrequests a file using internet protocols. The requested file isoriginally encoded using the techniques in this inventive material,particularly the iterative encoding technique using the spatial modelfor random-like data, which is usually the case for compressed data. Atthe client side, the encoded file is downloaded. The decoder may thendecompress a single file or multiple files using the tree data structurefor iterative decoding. Such a single file may be a photo from aninteractive media file, such as a website or application; the entire setof files being the whole website or application. Another use of thepresent method, article comprising machine instructions, and apparatususing databases is for cloud computing services, where a client machinecan download an application or use a hosted application over a network.The application can be encoded/decoded recursively, thereby facilitatinga fast download of the software for the client and enabling the user toopen the application. When the user closes the application, the clientcomputer may compress the software and store for later use, therebyminimizing storage space on the client computer. Techniques in the knownart that compress data can only encode data by a certain amount.Therefore, iterative data compression, as presented in this inventivematerial, can be of great benefit to databases when transferring filesover the Internet, leading to databases requiring less storage media,less data being transmitted over a network and less power consumed.

Data Modeling and Analytics: Data modeling is used for a wide range ofapplications when an analysis is needed for a particular type of data,such as weather, stock prices, and human behavioral statistics. Datamodeling assists with the planning and management of many differentservices that use data. Implementing data systems with the spatialstatistical model described in this inventive material to analyzesystemic characteristics is a very powerful way to analyze data withcomplex relationships. Spatial statistical modeling can find shapes andprobability field patterns for long sequences, which represents a datasystem's structure and other characteristics. The most and leastprobable data system can be deciphered using spaces of more than twodimensions. For example, statistics such as date of birth, height, andnationality can be discovered spatially using three state dimensionsalong a fourth interval dimension. Performing a spatial analysis of thedata system's structure, which in the preceding example includes therelationships between the three variables, can help to reveal astructured pattern to a lengthy time series, which may not be availableusing other methods in the known art.

Computer Generated Data: Computer generated data can be used for visualeffects in entertainment, medical imaging in health care, encryption insecurity, and physical simulations in various fields of science.Computer Generated data can require a considerable amount of processingand memory to generate. This is particularly true for constructingvirtual polygons and voxels. Where compression can be useful is in thearea of raytracing and polygonal collision. Raytracing is a techniquethat allows a processor to calculate light rays that reflect in adigital scene composed of polygons, which are typically triangles. Forphysical simulation such as fluids and hard/soft dynamics, the processormust calculate whether or not a polygon collides with other polygons inthe scene at a given interval. The system must store and accessextremely large amounts of data, which includes point locations of allthe polygons and/or voxels to render/simulate the scene. The method ofiterative encoding/decoding using the spatial statistical model toidentify the state subset for random-like data can be used to compressthe polygonal information in a scene, which can conserve memory. Forexample, the method would iteratively compress the polygonal data of thescene using a tree data structure that would be structured in a similarway to binary partitioning the scene, which allows a processor to findintersection hits between rays and polygons more efficiently. When a rayis fired into the scene, only the polygons that are near the ray wouldbe decompressed, conserving memory and processing resources. Thisdynamic access to the scene using iterative compression can also allow asystem to generate greater detail in the final rendering/simulationconsidering the memory limitations of a data processing device. Also,only the geometry or voxels seen by the camera can be decompressed,while other geometry hidden from view remains compressed.

An Information Generator may be created using the constraints of thespatial statistical model as guides for the generation of a data system.This is particularly useful for the generation of random-like data. Forexample, using the model as expressed in the preferred embodiments, amethod may use said models to generate monotonic sequences that onlyexist within the positive probability subspace and that have adistribution as indicated by the probability space using saidprobability distribution functions or maps. The method can then generatea sequence using the spatial model as pseudo-random or random data anduse such files for encryption to secure files.

Encryption: In addition to using the method, article comprising machineinstructions, and apparatus to generate a random-like file for use as anencryption key for data security purposes, out of the many possibilitiescontemplated, the manner in which the sequence function reads the datasystem may also be used to encrypt the data. For example, the sequencefunction can read units of the data system in as many different readpatterns that are possible given the size of the data system. Byprocessing the members of the ordered sequence in a variety of sequenceorders gives the method, article comprising machine instructions, andapparatus a way to encrypt the data further, as both the sequencefunction and the inverse of the sequence function are required toencode/decode the data. If the both the decoder knows which sequencefunction is used to process the data system and by which symbol setcriteria, then the decoder can decode the ordered list of identifiers.Otherwise, the ordered list of identifiers will be undecipherable andthe original data system unrecoverable. A file may be encrypted to aneven greater degree by implementing iterative encoding, where each cyclecan use a different reading pattern by a variety of sequence functions.

Media Transmission: Where a recursive encoding technique would be usefulis for a video or audio file. Various sections of the file may becompressed in a tree data structure, where each node references a pointin the recursion where a merging/appending of data systems occurs. Usingthis information, the decoder will decode only the blocks of data closeto where the video or audio is currently being played. The tree can bearranged to facilitate the partitioning of video sections. As the fileplays, sections of the stream that were played are recompressed, whilethe current and upcoming sections are decompressed as the pointer movesalong the video/audio. This is accomplished by finding the shortest pathin the tree between the child node signaled by a pointer (the currentframe or section/clip being played) and the root of the tree, which isthe final compressed file. The rest of the compressible data blocksoutside the shortest path are not decoded. This helps reduce resourcesof a data processing device by only decompressing played parts of thevideo/audio stream.

Images: As a possible example that may be applied to the present method,article comprising machine instructions, and apparatus, a threedimensional space may be used to model a video or picture; a pixelsystem. Out of the possibilities contemplated, this may be modeled usinga sphere, where its elements are also a three dimensional sphere thatuses three state dimensions. The states of the pixel may be representedin the following way; a yawing angle/dimension, which contains allpossible states for red, a pitch angle/dimension that contains allpossible states for green, and a depth dimension, which extends from thecenter of the sphere element to its outer shell, which contains all thepossible states for blue. Each unique state of the pixel is representedby an element of the sphere. A sequence of pixels and its possiblestates may then be summed using a monotonic sequence function, where asequence of spheres is contained within the larger sphere space. Themembers of the path of the three dimensional sequence leads to a placeand depth within the larger sphere space, comprising the smallerelemental spheres. An image or video with a unique systemiccharacteristic will have its monotonic path end at a unique location, asis the case for the two dimensional spaces described previously.

Media Acquisition: High definition video cameras are used extensively infilm and television media. 3D video is also used in these relatedfields, which requires twice the data to store. Compression methods areneeded for such media due to the large amounts of data usually acquiredfrom such devices. Using the techniques presented in this inventivematerial, a video camera device would record the media. A data processorwould read the incoming stream of bits as the recording commences,dividing the data into blocks for more efficient processing and compresssaid blocks. The resulting compressed data being information of highentropy, a specialized hardware implementation of the iterative dataencoding method by a data processor may then take said blocks andrecursively encode the data using the spatial statistical model forrandom-like characteristics. Because the data is primarily for storage,a tree data structure for the iterative encoding may not be required.This data may then be stored at the camera using a storage drive orremotely using fiber optic means and RAID storage.

Pattern Recognition: One of ordinary skill in the art may discern thatthe same aspects described in this inventive material to recognizerandom data can be applied for the recognition of ordered data, such astext, speech, imagery, movements and other observables. Indeed, the sameprocesses presented in this inventive subject matter can be implementedto recognize any incoming signal. Patterns can be recognized within aspace along the interval dimension (time), which may be any kind ofpattern based on a variety of symbols; i.e., letters, shapes, sounds,etc; whereby the possible elements within a space are enclosed byprobability constraints or boundaries. A match is made when the valuesof the features (members) are within a particular state subset, which isdefined using probability constraints, probability distributionfunctions and probability maps, as explained in the previous exampleembodiments.

For example, the data processing system reads an incoming stream ofphonemes (observable features). It then plots the sequence of phonemesas members of an output sequence that correspond to elements of a space,given particular symbol criteria, by applying the methods described inthe first and second embodiments. A function may use “curve fitting” tospecify a boundary defined by a continuous function, which bound theelements that correspond to members of the output sequence. Acalculation test may be initiated using a library of other models storedin a storage medium to see whether or not the current boundary “fits,”matches, or is enclosed within the constraints of another boundarymodel. One skilled in the art may use a straightforward approach forthis test, such as an intersection test based on the elements of thespace, described in the first example embodiment.

A match may also be made using techniques described in the third exampleembodiment, using probability distribution functions and probabilitymappings to model the probable states of data systems within the space.With regards to pattern recognition, the method can select states orsequences that correspond to elements associated with a higherprobability than other elements. The method then generates the generatedsequence where its members corresponds to the most likely state. Forexample, the data processor receives a sequence of phonemes. The dataprocessor will receive, from a storage medium, the spatial model, whichindicates the most probable sequence of letters based on the probabilityfield established by the probability distribution and mapping functions.The data processor will then generate a sequence of letters that followsthe most probable sequence of the spatial model.

Out of the possibilities contemplated, another possible embodiment forpattern recognition will use a multidimensional space. For example, thehue and brightness values of the pixels from an image may be structuredusing a trigonometric sequence function that processes the orderedsequence of pixels, whereby the output sequence coils to form a tubularshape, which was described as a possible embodiment of a sequencefunction in the first example. The values of hue and brightness distortthe length of the radius for each member as the path is swept from thecenter of the circle. The result is a tubular pattern similar toindentations on a surface of a tube. These indentations may be used as afingerprint for the data, which may be compared to other data systemsand models. The algorithm for matching would use the same techniquesdescribed in the first example, particularly using a collision detectiontype algorithm, where the elements are treated as voxels. A match may bedetermined by the data processor based on how close or far the elementsof the boundary are from the spatial model being compared. For example,if the boundary intersects too deep with or is spaced too far from thespatial model, the processor can determine that a match may not be veryprobable.

The steps outlined for pattern recognition are not a limitation of thepresent inventive subject matter. Other steps may be employed using keyaspects of the inventive material for the purposes of patternrecognition. This is merely an example of how the key aspects of theinventive subject matter may be employed for the purposes of patternrecognition.

It should be apparent to those skilled in the art that many moremodifications besides those already described are possible withoutdeparting from the inventive concepts herein. The inventive subjectmatter, therefore, is not to be restricted except in the scope of theappended claims. Moreover, in interpreting both the specification andthe claims, all terms should be interpreted in the broadest possiblemanner consistent with the context. In particular, the terms “comprises”and “comprising” should be interpreted as referring to elements,components, or steps in a non-exclusive manner, indicating that thereferenced elements, components, or steps may be present, or utilized,or combined with other elements, components, or steps that are notexpressly referenced. Where the specification claims refers to at leastone of something selected from the group consisting of A, B, C . . . andN, the text should be interpreted as requiring only one element from thegroup, not A plus N, or B plus N, etc.

What is claimed is:
 1. A system comprising: a processing device; a non-transitory, tangible medium that stores instructions that when executed by the processing device cause the system to perform operations comprising: accessing a value corresponding to a point comprised by a space of a spatial probability model, the point corresponding to a state comprised by a state superset of at least one of the following: a data system or a systemic characteristic of the data system; determining an output based at least in part on the value corresponding to the point comprised by the space of the spatial probability model, the output corresponding to at least one of the following: a boundary of a state subset, the state subset comprising a subset of states of the state superset, or a probability of the state comprised by the state superset of: the data system or the systemic characteristic of the data system; using the output corresponding to at least one of the state subset or the probability of the state, to construct data with reduced redundancy and/or reduced average information entropy; using the data with reduced redundancy and/or reduced average information entropy to perform at least one of the following: generating and transmitting a message comprising the data with reduced redundancy and/or reduced average information entropy, or generating and storing a computer file comprising the data with reduced redundancy and/or reduced average information entropy.
 2. The system of claim 1, the operations further comprising: receiving the data system; using a sequence function that operates on an ordered sequence of data units from the data system to generate an output sequence; corresponding the output sequence to the point comprised by the space of the spatial probability model by employing a function that at least: determines a number of times the state, comprised by the state superset of the data system or state superset of the systemic characteristic of the data system, corresponds to the point comprised by the space, or determines a number of times data units of the output sequence correspond to the point comprised by the space, or determines a probability of the state, comprised by the state superset of the data system or the state superset of the systemic characteristic of the data system, corresponding to the point comprised by the space, or determines a range of probable sums of the ordered sequence of data units from the data system, or determines a probability distribution indicating the probability of the state, comprised by the state superset of the data system or comprised by the state superset of the systemic characteristic of the data system, corresponding to the point comprised by the space, or determines an identifier representing a state subset type, or determines an identifier enabling an assignment of the probability of the state, comprised by the state superset of the data system or comprised by the state superset of the systemic characteristic of the data system, corresponding to the point comprised by the space.
 3. The system of claim 1, the operations further comprising supercorresponding at least two states of the state superset to a same point comprised by the space of the spatial probability model.
 4. The system of claim 1, wherein the spatial probability model comprises at least a Euclidean space, a Non-Euclidean space, a topological space, or a phase space.
 5. The system of claim 1, wherein the space of the spatial probability model comprises at least two dimensions.
 6. The system of claim 12, the method further comprising implementing, by the data processor, a hierarchical multi-resolution structure for the space and/or the output sequence.
 7. The system of claim 1, the operations further comprising determining a location of the point comprised by the space in a continuous space.
 8. The system of claim 12, wherein the sequence function comprises at least a mathematical monotonically increasing function, a summation function, a multiplication function, a delta function, a trigonometric function, a lexicographic ordering function, a sorting function, or a sequence function that generates a delta value of the output sequence that is a measure of a variance from a determined value.
 9. The system of claim 12, wherein the ordered sequence of data elements includes at least sixty-four bits.
 10. The system of claim 1, wherein the systemic characteristic of the data system corresponds to a structure, a unit of data, a data subsystem, a relationship between units or subsystems, a process, or a spatial form.
 11. A method comprising: accessing, by a data processor comprising hardware, a value corresponding to a point comprised by a space of a spatial probability model, the point corresponding to a state comprised by a state superset of at least one of the following: a data system or a systemic characteristic of the data system; determining, by the data processor, an output based at least in part on the value corresponding to the point comprised by the space of the spatial probability model, the output corresponding to at least one of the following: a boundary of a state subset, the state subset comprising a subset of states of the state superset, or a probability of the state comprised by the state superset of: the data system or the systemic characteristic of the data system; using the output corresponding to at least one of the boundary of the state subset or the probability of the state, to construct, by the data processor, data with reduced redundancy and/or reduced average information entropy; using the data with reduced redundancy and/or reduced average information entropy to perform at least one of the following: generating and transmitting a message comprising the data with reduced redundancy and/or reduced average information entropy, or generating and storing a computer file comprising the data with reduced redundancy and/or reduced average information entropy, or generating an analysis of the data with reduced redundancy and/or reduced average information entropy.
 12. The method of claim 11, further comprising: receiving, by the data processor, the data system; using, by the data processor, a sequence function that operates on an ordered sequence of data units from the data system to generate an output sequence; corresponding, by the data processor, the output sequence to an element of the spatial probability model by employing a function that at least: determines a number of times the state, comprised by the state superset of the data system or state superset of the systemic characteristic of the data system, corresponds to the point comprised by the space, or determines a number of times data units of the output sequence correspond to the point comprised by the space, or determines a probability of the state, comprised by the state superset of the data system or the state superset of the systemic characteristic of the data system, corresponding to the point comprised by the space, or determines a range of probable sums of the ordered sequence of data units from the data system, or determines a probability distribution indicating the probability of the state, comprised by the state superset of the data system or comprised by the state superset of the systemic characteristic of the data system, corresponding to the point comprised by the space, or determines an identifier representing a state subset type, or determines an identifier enabling an assignment of the probability of the state, comprised by the state superset of the data system or comprised by the state superset of the systemic characteristic of the data system, corresponding to the point comprised by the space.
 13. The method of claim 11, the method further comprising supercorresponding, by the data processor, at least two states of the state superset to a same point comprised by the space of the spatial probability model.
 14. The method of claim 11, wherein the spatial probability model comprises at least a Euclidean space, a Non-Euclidean space, a topological space, or a phase space.
 15. The method of claim 11, wherein the space of the spatial probability model comprises at least two dimensions.
 16. The method of claim 12, the method further comprising implementing, by the data processor, a hierarchical multi-resolution structure for the space and/or the output sequence.
 17. The method of claim 12, wherein the sequence function comprises at least a mathematical monotonically increasing function, a summation function, a multiplication function, a delta function, a trigonometric function, a lexicographic ordering function, a sorting function, or a sequence function that generates a delta value of the output sequence that is a measure of a variance from a determined value.
 18. The method of claim 12, wherein the ordered sequence of data elements includes at least sixty-four bits.
 19. The method of claim 11, the method further comprising determining a location of the point comprised by the space in a continuous space.
 20. The method of claim 11, wherein the systemic characteristic of the data system corresponds to a structure, a unit of data, a data subsystem, a relationship between units or subsystems, a process, or a spatial form. 